A blog for sharing my math interests on the web, to post new materials for elementary, secondary and teacher ed, and vent mathematical steam when needed.
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I was inspired to do this by the neat quadrilateral hierarchy sketch shared this morning on Twitter. But I got wishing they had made the types accurate - that you could only make squares in the square spot. And that their hierarchy used the inclusive definition of trapezoid. (Pet peeve of mine.) Then I thought what if the types lit up when you make the shape? That led to the sketch pictured below, available as a ggb file (EDIT: in GGB 4!) or as a webpage.
When I included it as an applet here, it just didn't work as smoothly as it does over at the geogebra hosting, or by displaying the file directly in a browser.
It was very fun figuring out the conditional tags to make the names show up. I think I've covered most of the corner cases. Figuring out a way to do convex/concave and quadrilateral or not
It was quite handy knowing multiple definitions of each type, which me wonder about a scaled down version of this as a problem to assign to students.
Where do you stand on the trapezoid definition? Is a parallelogram a trapezoid? (I say yes!)
If at some point in this post you don't say "that's a bit of a stretch," I will not have done my job here.
I got an iPod Touch for work this summer. Despite being aggressively pro-tech, my personal tech level is loooow. No cell phone, no iPod, no iPad, no video game system ... ridiculous, really. But I'm working with Alejandro Montoya, a computer science grad student, this summer (following my colleague Char Beckmann) as he develops a cool iPhone quadratics game. (Due in the app store for free any day now.) One of the problems was a ridiculously low number of devices, so, time to invest. My colleague Paul Yu and I have an NSF proposal in to equip a classroom with iPod Touches (among other tech) and I'm a believer in really using things before asking students to do so. Paul and Dave Coffey use their iPhones well to support their class.
While developing the game, Jon Engelsma, director of my university's mobile development lab, wanted the game to use more of the iPhone specific capability. Each phone/pod is equipped with an accelerometer - which is why it can detect orientation and movement like tilting. In response, Alejo added a new aspect to the game where you're making the parabola in the air to show orientation. But along the way, Jon mentioned Doodle Jump as an example of a game that used it well. "Oh, only 99¢," I say.
Ruh roh.
At the time of writing this, I'm at a high score of about 20,000. I swear I don't use real time to play, just moments where I'm stuck somewhere.
There's math in that game?
My son asked that in surprise. As Alejo mentioned to some of the high school students playing ParabolaX, there's a LOT of math in programming and game design. Frank Noschese has written so much good stuff on the physics of Angry Birds. Doodle Jump has that. I've thought about how far up and across you can get on a jump, use constant speed estimation of moving platforms, etc. But other than doing some modeling (which would be interesting I think) of the jumps, there's not a lot of explicit math for players. There is, though, an understanding that the game is on a cylinder (go off the left, return on the right, etc.). That's made me wonder if anyone has developed an iPhone game that really uses the accelerometer to explore topologically interesting surfaces. A Möbius maze that you navigate with the accelerometer? A Klein bottle version of Othello? Who knows where that could lead.
There's math in that game
The Common Core State Standards Mathematical Practices:
1. Make sense of problems and persevere in solving them.
At its heart, like many games, Doodle Jump is a big problem (get as high as possible) composed of smaller problems (how do I get past the pink monster reliably). My perseverence, like most students, is very high for games.
2. Reason abstractly and quantitatively.
The constraints are simple: you can't stop jumping, and if you land on nothing or bump into an obstacle, game over. There's not much quantitative reasoning, unless you're fixated on a score. Then there's some nice linear programming on the fly to figure out how to improve your score and what's required. I have wondered why, being extremely right handed, I seem to be better with my left on this game. Specific game questions too, like why don't I see rocket packs anymore?
3. Construct viable arguments and critique the reasoning of others.
Hasn't happened for me yet, because I haven't discussed the game with anyone other than my son, who's even more of a novice than I am. But there's definitely opportunity, as we've discussed: Does a back lean enable higher jumps? That's my son's conjecture but I haven't experienced it yet.
4. Model with mathematics.
Also not yet. Though I am interested in trying to model the jumps, and I'm curious about how to even start. There's good reason to measure to improve your game play, but the modeling is the kind of idle mathematical reasoning that mathematicians love. I'd also love to know how the accelerometer data feeds into the shape of the jump. Constant horizontal velocity if tilted or does it depend on the angle of tilt?
5. Use appropriate tools strategically.
Not yet. But the modeling will definitely be helped by mathematical tools.
6. Attend to precision.
Hmmmm. Kinesthetic precision is definitely required. It matters how you move and how much. The engagement of this again has me wondering how to make math more kinesthetic more often.
7. Look for and make use of structure.
Very rich context for this. Even at my lower levels, there are many patterns to notice, and noticing them is crucial to survival in the game.
8. Look for and express regularity in repeated reasoning.
I think it's a new thought to me that this is a crucial part of video games, and I want to use this connection in math class. The skills required early on become automatic and no longer a problem. This is just natural as you get better, you gain automaticity with tasks that used to require thought. Even though you may make occasional mistakes. Does that describe math or video games?
What am I learning about teaching?
Half of learning about teaching is learning about learning. This game has been good for me to think about because I'm not very good. There was a brilliant teacher educator at Siena Heights, Sr. Eileen Rice, required her advisees to take a class in a subject with which they struggled. Great idea. Just like with a math problem, we can't problem solve if it's too easy for us. It's struggle that affords an opportunity for growth.
In the game I've had to figure out specific challenges, wonder about how things worked in general, identify areas where I need improvement, and make some realizations about my limitations. I can not shoot effectively. My videogame dexterity and reaction time is probably below average. But I've had nice moments of achievement. Figuring out how to get past a few beasts, then that I can jump on them, specifically working on using the cylindrical aspect, etc. Things I couldn't do before that I can do now.
The other half of learning about teaching is tackling the question of how do I support learners? That's the heart of instruction to me, and what motivates gathering data (assessment), giving feedback (evaluation) and problem and resource selection (planning). This game has made me wonder about all these.
Do I look up directions? Cheats? The game is popular and there are lots of tip sites out there. Even Doodle Jump cheats. This requires me to think about my purposes. Do I need the highest possible score? Do I want to figure it out for myself? What's the purpose of playing? Once I was a teacher that told students this information without asking. I was good at it, and got good evaluations, and most of my students did really well on tests. Then I was a teacher that would never tell this information, even if students basically begged for it. Students did some amazing work, found out things I hadn't known, and most were successful. But some students were frustrated, including some of the successful students. Now, I try to assess student purposes and provide relevant information. But it's harder than having a simple extreme policy.
Wow, I made it through that whole paragraph without mentioning how Khan Academy can be like those tips and cheats YouTube videos. (Shoot.)
Talk. It's also better with other people. I've been hampered by doing this game alone. Looking at the practices made me think about the richness I'm missing. I've seen this in Alejo's pilots of ParabolaX also - radical differences in what students get out of the game based on how much they discuss it. I want a healthy balance for my students between 'let me do it for myself' and 'how are you thinking about it?'
How do I measure success? I like the score as a measure of how far I've gotten, or as a measure of whether I've gotten better. But as I consider that, there's really better things to notice. The game territory changes as you climb, so that's a good measure of how far. (When I get to the bounce-once platforms currently, that's a good game! In the jungle setting.) Can I make difficult moves? Have I gotten better at getting better? It feels like I have better learned how to identify problems in the game and am more efficient at addressing them.
Then I got wondering what the scores even really mean. Is my high score the best measure of how good a player I am? Should it be my average or median score? Weighted somehow between the two? If the goal is the maximum score, it encourages high risk behavior that's bad on the average but when it works garners big points. Score-based thinking also pushes me towards tips and cheats, which are not in my best interest as a learner or enjoyer of the game. The best use of the scores is a nice mathematical problem, like a simpler version of trying to figure out what are the relevant baseball statistics. In general, though, it encourages me to go farther in the direction of SBG and portfolios.
So...
If you made it this far, thanks for sticking with the rambling. I'd love to know what you think about this, or other thoughts you have about teaching and learning from games or other odd contexts.
P.S. Yes the title is a little poke at the now ubiquitous Jump Math, which has some super-proponents. (There are samples at jumpmath1.org)
P.P.S. We need some games this engaging with more math content. Waker and MangaHigh are a start, but the accelerometer using games will be a big step up.
Image credits: there were no good CC images for this post, so if I used one of yours and you wish it not, just let me know. All images click through to original source. The ragecomic was too true not to include.
In general with technology, I think there are few must haves. I much prefer tech that is usable in pieces rather than by wholesale adoption. For example smartphone use vs TI-Inspire. This is written with iPad in mind, but would apply to any tablet computer, I think.
iPads and tablets seem like that kind of technology. For my school several of us were asked to think about how teachers could use iPads. Most of the uses involve students having the iPad. So I thought I would put down what I think of, and then troll for other suggestions in comments. I'm looking for ideas that involve the teacher bringing one iPad into the room. Also if there are ideas to distinguish it from a smartphone or laptop.
Ideas
Give it to students. Do they need to look something up, compute something, watch a video, capture a video, read a reference text... The larger interface makes it more suitable for group work.
Constructive use of social media in the classroom. Backchannel, Google Groups or Plus, Edmodo, etc. Recently did #mathchat on Twitter with a class and it was a drag bopping back and forth to the classroom computer and taking dictation on class comments. Instead I could have said, "here's the iPad."
Document your experiences to share with students. My colleagues Dave Coffey and Sean Lancaster do this already. Twitter, Evernote and I suppose now Google+. Portability > laptop, interface > smartphone.
Data collection in class. Check off attendance, notes on student work or participation. Hard to carrry your laptop around to each group, easy to carry iPad. I used to so some of this on a palm but it was quite clunky. Apps for this are developing rapidly. BlackBoard Mobile is being pushed hard and may figure in for GVSU.
Assistive technology. Differentiate your lessons for a student with distinct requirements for access to a lesson. (Spellcheck suggested Assertive Technology - I like that, too.) See for example this post on iPad assistance (via @langwitches on Twitter).
Moves towards paperless workplace. All those meetings where there's handouts for each teacher that move immediately to the recycle bin...
Meta-use: model new technology adoption and integration for your students. By trying out new tech and sharing your process, you are modeling towards their Technological-Pedagogical-Content-Knowledge development (for teachers; see http://www.tpack.org) or the similar structure in other fields.
Who knows? Apps are being developed at a break-neck pace. Putting these devices in teachers hands will expand their capabilities in ways we don't even know because they being created tomorrow. Cf http://www.mobilemouse.com/ that will allow you or students to control the classroom computer from anywhere in the room. There's a dedicated iTunes room for teacher apps now. (Warning iTunes link opens in iTunes.) Quite an opportunity for innovation in scholarship of teaching or even collaboration with the University's own mobile development lab. Or for students who are budding developers.
We just finished up our Vacation Bible School, which had Joshua as a theme. I'm the storyteller, and my thoughts were much on teaching throughout the whole process. Dave Coffey and I often talk about teaching as story telling (EDIT: turns out he was writing about storytelling at the same time as I was - shocking), and love drawing lessons from Harry Potter as we both love the books. (So much so that when my otherwise-entirely-admirable summer class revealed they had not read them I was at a bit of a loss when it came to examples for questioning in literacy. Stunned, I was.)
The process started with learning about the subject. I'm not a bible scholar, but I am willing to write bible studies. Mostly I just share the questions I wonder about as I try to make sense of the readings. I use BibleGateway.com as a tool, as it has many translations and a robust search feature. I copied the relevant bits of scripture out, and made it into a study. (Shared here.) I got to discuss it with three groups of people before writing the story. In particular, with a men's group where (rather unbelievably) I'm a junior member. Finally it was time to write the story. (Shared here.) I wrote the narrative, following the facts of the story. I went over it again, thinking about the teaching point(s) of each of the three days. I'm not always explicit with those, but it's going to be harder for kids to notice if I don't put it in there to notice. I went over it again, strengthening and adding connections. Things like: if the ark is going to be used on day 2, make sure it's mentioned where I can on day 1. Go over it again - does it include what's important and is what's important emphasized? Finally, practice the telling.
My daughter performed with me to help with the illustration and comic relief, and we'd go through the story right before telling it. Originally she was going to be Joshua the first night when young, then Israelites later, but we wound up keeping her as Joshua because the kids seemed to identify her really strongly with the part. As we told the story, we responded to the kids' response. We monitored both their general engagement and asked questions about what they would do, or what thought might happen or what they knew about things in the story. We told one version to the K-4 crowd, and another to the age 3 and 4 group (with lots more marching around). We used the props and set pieces we had and figured out how to do things with limited time and resources.
Sounds a lot like teaching, right?
I've been really interested as some teachers share on Twitter their summer planning process. It seems some are frustrated by trying to plan without their students, and I agree that this is questionable. But I also got to thinking that it offers an opportunity to think about your objectives in another way. As a story. Your plans won't be able to be set until you know your students, and assess what they're bringing to the story. But you can think about what's important and look for the narrative. To me it's quite like what Dan Meyer's been writing about when he's considering how to set, describe and pitch the problems you find. I also think that's the kind of work we need to find better and better ways to share with each other, as it's very portable amongst schools and students.
I had a great experience this summer co-leading with Michelle Bunton a GeoGebra workshop for teachers. We decided to do a loose structure, emphasizing algebra one day and geometry the second. People were free to register for one or both days. We got about 22 teachers, with 14 for both days. One teacher was bitterly disappointed because they wanted premade activities, and our focus was on learning the program. Though we tried to connect people to plenty of resources and the wide-world of GeoGebra sketches, this teacher left before that. Most of the teachers took the freedom and ran, and it reminded me of what a joy it is to be in a room full of independent, motivated learners working on stuff that matters to them. I learned a lot, which is typical of such situations.
We decided to run the workshop using GeoGebra 4, as it's being released at the end of the summer. Unfortunately - I was a novice on it. I'm an enthusiastic GGB 3 user, but have been weak on spreadsheet use. Adding more novel features was a little scary. Turned out well, as it made us co-investigators with the teachers. And the program is great. I mean it was great, but now it is greater. They've managed to add features without making it perceptibly more complex. That's rare; I love this software and its developers. Guillermo Bautista's GGB4 sneak peek series was very helpful. (Of course!)
We structured the workshops with time to get software loaded. And I wanted the teachers on Twitter to converse on backchannel through the day. That was hit or miss, due to Twitter's tendency to ignore new users as an anti-bot measure. I wound up having people follow me so I could follow them, then retweeting their tweets. This made some people show up but not others... mystery. I thought it was important because I get some of my best GeoGebra support on Twitter. And, in fact, during the workshops we got some excellent input from @mike_geogebra and @lfahlberg .
So startup, a demonstration sketch to show some of the potential for classroom use, an overview of the program parts (tool bar, menus, views, etc.), specific tasks to figure out how to use, and then free explore time. The workshop website has all of our materials (see the workshop page), plus the sketches created by participants. One exciting feature is that participants spent time sorting and classifying sketches by standards strand in a Google spreadsheet with links to the sketches.
One problem were still working on is the "now what?" We're trying a Google group and want the website to morph into a place for teachers to share sketches. (If you are interested in joining the page to share your materials, just drop me a line.) Teachers also wondered aloud what made this experience useful in comparison to some other professional development, and that's worth looking into.
As I said, I learned a lot: about professional development, workshop development with a new partner, and a lot of GeoGebra. The GeoGebra knowledge I could put into a sketch! Note that the sketch links below use GGB4, which can download as a Beta.
Dave made a nice height vs time projectile sketch that used text box inputs. (That is my favorite new feature so far.) Cristine got me thinking about more subtle show/hide condition with her beautiful Unit Circle. One of the participants made a face sketch for which he figured out how to limit the domain of a function. (Which he seems to have never uploaded!) @lfahlberg taught us how to make a reset button for a sketch. Chris and Jason just pushed and pushed and explored in general. I can't remember why we figured out that sliders can have calculated min and max now.... I was seriously impressed at what good use everyone made of their time.
After the workshop I made this projectile sketch to practice. It's a projectile in the x-y plane, where the slider advances or animates time. You put in the initial height by textbox, set the vector of your throw by dragging the vector, and can do target practice to a garbage can or by throwing at a seagull. (No actual seagulls were harmed yotta, yotta.) The button resets the targets. This was fun to make.
The big screen shot is an animated gif - one of GGB's new export formats. Go ahead and click on it! The garbage can was inspired by Dan Meyer's WCYDWT ball toss, which is also a sketch.
GeoGebra is a supertool, which even has the power to get more powers, and I definitely want my students and preservice teachers skilled in its use.
I've been loving the posts people are writing for this year's virtual conference responding to the prompt: What is at the center of your classroom?. The conference is an amazing collection of writing put together by Riley Lark, blooger at Point of Inflection. (Which I've always liked because it sounds - and reads - like he's synthesizing intention and reflection.)
This is an attempt to emulate my betters, which is actually not a bad strategy for improvement.
I want the center of my classroom to be empowerment. As a bad beginning teacher, emulating David Letterman of all people, I realized that I loved teaching math. I'd tell people that there weren't many things you could teach where the student would literally be able to do something at the end of the class that they couldn't do at the beginning. While I still like that, I now think it can happen in many more places than math class, and have a much different idea of what I want the students to be able to do.
On my own I got to realizing that problem solving was what I really wanted to teach, and my friend Sue Feeley introduced me to Polya (figuratively) and the other NCTM process standards. Both helped give me language to describe things I had realized, and both indicated a path to set out on. Vygotsky helped me understand why students responded so differently to the same task.
Mosaic of Thought, Cambourne's Conditions of Learning and other literacy education work helped me start to understand how to teach processes, and I don't know that I would ever have found that were it not for Kathy and Dave Coffey. The conditions are the heart of what I want for my students, and creating or nurturing those conditions is what I see as the main responsibility of my work.
Engagement is first and foremost. Cambourne notes that engagement requires learners to believe:
they are potential doers of what is being taught. This fits with and explains the idea that the students need to be the ones working in the classroom. (See also expectations)
what they are doing matters to them. This is why teaching the processes is so important to me. Polynomial division does not matter to 99% of students. Problem solving will make 100% of my students' lives better.
they are safe to try. This is why classroom community is so important. (See also approximation.)
Cambourne links engagement with two conditions:
Immersion. Learners need to experience real and rich mathematics of all kinds. Still one of my measures of how rich a question is is to consider how many processes it invites learners to engage in.
Demonstration. Learners need many and authentic demonstrations of what they are learning. This was huge for me. I had become a hardcore discovery-based teacher. I was proud of my students saying things like "Why ask him, you know he won't tell us." (Oh, that is painful to me now.) I took students' feedback about their frustration and decided that it was because they weren't used to this mode of learning. But really, I was asking them to do things they'd never seen. Asking them to learn how to dance when they'd never seen one. Coupled with a lack of good feedback, it's a real testament to those learners that they got as much out of it as they did. Adding demonstration let me back into the discussion. Not to tell the learners how to do it, but to share with them authentically how I think about it. I make space for them first, as I prefer if they're demonstrating to each other, but I watch and assess for when they need demonstration as a support.
The conditions that make engagement more likely are:
Expectations. This goes hand in hand with the Equity Principle from the NCTM, which is near and dear to me. I believe all people can do significant, important mathematics. I really do. I try hard to communicate this to my learners.
Responsibility. Learners make their own decisions about when, how and what to learn. THIS IS ANARCHY. I know. It's dangerous, especially when our learners have been trained for dependence and helplessness. Most students are not ready for full freedom immediately, but it is my goal for all of them. It's also my ongoing struggle to get students to understand that I both mean this and it doesn't mean that work is optional.
Employment. Learners need time to try out their learning in authentic situations. This connects with real life mathematics, with project based learning, with discovery learning and more. The students need to be the ones working if they are to be the ones learning.
Approximation. Learners need to be able to make mistakes without fear of punishment. If there's one area that math has completely screwed up on in the past, it is this. I do it, now you do it perfectly. This is crazy. We all know that no one learns anything important this way. And that the mistakes people make are crucial for learning in the first place.
Response. Learners need real and meaningful feedback about what they are interested in working on. I now have my students put stickies on any work they turn in with what questions they have for me. Cambourne: "response must be relevant, appropriate, timely, readily available and nonthreatening." Grades are not feedback in this sense, and can only strive to be timely and readily available. (Although that does make grades better so far as it goes.)
Though I have made progress in creating the conditions, I have a long way to go. (Although that implies an end to the journey and there may not be one.) They help determine what I teach, what my classroom is like, what I want to know about the students and their learning and how I evaluate their work. That I would not have found them without colleagues in my professional learning community speaks to the heart of why that is important. So thank you, to Sue, Dave and Kathy, for helping me get this far.
So many things to write about to catch up... but it's been a while since I posted a game, so with an impending Math Teachers at Play at Math Mama Writes. The submission form seems to be wonky, so submit directly to Sue. Plus this kind of fits with U.S. Independece Day, as we have been known to fight a war or two.
This game is good already, but could be great. So if you have feedback, let me know, please! As is, it's probably best used as a review game, but I'll comment afterward about how it could be used as a framework for a unit.
Set up: Make your own deck: 11 lines. Each line should be drawn so that it passes through at least two points with integer coordinates, such as (-2,4) or (5,5).
Claim your deck! Mark each line card on the graph side with your insignia. Initials, emoticon, math symbol, etc. – your choice. Tip: make your cards NICE and personalized. Decorations and alterations that do not obscure the line or the math are not only permitted but encouraged.
War: 2-4 players. Each player needs a deck of 11 face down cards, shuffled or not – it’s up to you. Set aside any extras, make one more if you need it.
Players roll the die for the combat. (2nd roll and beyond, the winner of last battle rolls.) Flip over the top card of your deck and follow the combat rule. On a tie, flip over one more card to determine the winner of the battle. If more than two are playing, this is only on ties for best and only the people who are tied.
Play through the deck once. The winner is the player at the end with the most cards. Give cards back to the owner. Except for the Spoils of War.
Spoils of War: Out of the cards the winner captured, they take one card from the opponent’s deck to keep. Add your mark and cross out theirs. This may mean the loser needs to make a new card for their deck for the next game.
Example: The first roll is a 2. Least slope. -2 < 1
(The X and lemniscate are the players' personal marks.)
Math notes:
Use the cards for sorting activities before playing.
Have players keep track of hard to determine battles.
Discuss card design strategies.
What about undefined and zero slope lines?
What other combat rules could you have?
Handouts: as a Scribd file, and the graphs template. At the right is an image (larger when you click on it) that you could also print for the graphs.
Discussion: Ted, one of the excellent summer grad student/teachers, tried this cold as an end of year activity with a small group, and they struggled with it. He felt like it had a lot of promise, but that the math requirements kept students from the game since they were rusty with it.
Trying this with teachers convinced me of it's potential, as it even uncovered math for them to discuss, and generated situations they had to think about.
I could see this game being at the end of a linear unit, where students have been generating graphs as examples as they go through the topics, using them for activities like finding slope, sorting from least to greatest x-intercept, y-intercept and slope. Use them to construct tables or find equations. Non-contextual, but strong on representation. What do you think?
I'm trying hard not to use too many unlicensed images but this is too perfect. God bless you Bill Waterson, wherever you are.
Then I got wondering what the scores even really mean. Is my high score the best measure of how good a player I am? Should it be my average or median score? Weighted somehow between the two? If the goal is the maximum score, it encourages high risk behavior that's bad on the average but when it works garners big points. Score-based thinking also pushes me towards tips and cheats, which are not in my best interest as a learner or enjoyer of the game. The best use of the scores is a nice mathematical problem, like a simpler version of trying to figure out what are the relevant baseball statistics. In general, though, it encourages me to go farther in the direction of SBG and portfolios.
Handouts: as a Scribd file, and the graphs template. At the right is an image (larger when you click on it) that you could also print for the graphs.
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