Math Hombre

It should come as no surprise to anyone who spends more than 5 minutes on this site that I see a lot of connections among math, games and art.

My favorite game of all time is Magic: the Gathering. I love it in concept and in play. Amazing strategy, accessible at several levels and varieties and terrific flavor and art for bonus. But I'm not trying to convince you to try the cardboard crack - unless you're interested? - I'm just letting you know what I'm about to try to riff on. Mark Rosewater is the long time lead designer on MtG, and is very generous at sharing his design thinking, on Tumblr, in longer form blogposts and in podcast.  He is a serious student of game design, and focused on engaging play, so there are often connections to teaching and learning. A recent article is on narrative equity. One of the ways games engage players is the opportunity to make a story. It's a rich payoff, and can be significant to identity. I'd encourage you to read his post, but the examples after the intro story are in terms of Magic, so may not be accessible. Mark wraps up the introductory stories about his daughter and himself with this:

What do these two stories have in common? In both, Rachel and I prioritized having an experience. Our personal story carried enough value that it influenced how we behaved. It was an interesting concept, that people will give weight to choices based upon the ability to later tell a story about it. I call this idea "narrative equity."

The next step for me was applying this idea to game design. What does narrative equity mean to a game? Well, games are built to create experiences. I talk all the time about trying to tap into emotional resonance and capture a sense of fun. Narrative equity should be one of the tools available to a game designer to do this.

After thinking it through, I came up with seven things a game designer can do to help maximize narrative equity in their game.

What follows here is his list of game design connections to this idea, and why I felt like he was talking about teaching mathematics.

#1 – Create components with enough flexibility that players can use them in unintended ways

Math, to me, is ultimately about doing. We often make it about acquiring facts and techniques, and can lose track of why we are asking learners to do that. When learners are exploring these ideas, these powerful, culture changing ideas, which we are teaching, there are going to be ways to combine them to get new places. When we front load mathematical ideas, so that in the next section we can use them to solve this kind of problem, we're working against this.

The big shift for me on this was going from that linear learning curriculum model to a landscape approach like those in the Fosnot & Dolk work. (Image from this workshop.) They create a distinction among models, strategies and ideas, and realize there is a progression, but there are so many paths that learners can take from place to place. Formally or mentally, this is how I see curriculum now.

#2 – Create open-ended components that can be mixed and matched in unforeseen ways

To some extent, for me in math, this is about tools and representations. I am a deep believer that learners being able to represent (in the old NCTM process standard sense; create, move among and choose representations) magnifying their problem solving capacity. Given the ability to create graphs, diagrams, written/verbal descriptions, contexts, tables, equations or expressions... that creates excitement. I cannot tell you how often I learn something new or see a new idea and need to make it in GeoGebra or Desmos. And am delighted by the result. Or to write down a function to model a behavior. Or see a pattern in a table that was hidden from other perspectives...

Naturally, this dovetails with tool use. We live in the future, people, with free tech that gives capacities to everyone once reserved for super-geniuses. To some extent, I think why I stood out as a young math student was that I could do that in my head. Now everyone can! Why hide it from learners? Several of my Calc 2 students this summer had Calc 1 with NO TECH.  Augh! On the flipside, I felt like learning Desmos, GeoGebra and Wolfram|Alpha was a goal in my course, and was frequently happy to see them used in ways that we had not done. A good sign the learner is making it a tool of their own. We also had programmers making things, and a student from another U sharing his Mathematica programming, which they are required to use.

#3 – Design in unbounded challenges that allow the ability to create memorable moments

THIS. I want to get much better at this. The twist is that math does this naturally, so we've had to contort it to hide that aspect. I ask students to do this, but don't know how to support them. Especially when I see them, they have learned that the teacher always has an end in mind. Show us how! Show us an example! There are, of course, times for this. But when I ask you to see what you make with this, I really want to leave the door open.

I hit a pretty good middle ground with the quarter the cross assignment in Calc 2 this summer. We used David Butler's examples to launch it and model, but then opened the calculus door by connecting to how we had great area calculating power. Many exciting results. Not all of them, but I don't think we can require creativity. Just make space for it, and celebrate it. For the assignment, we had a little experimentation in class, a bit more in the takehome and then a lot for the people who chose that for a writing assignment.

#4 – Create near-impossible challenges that can become a badge of honor

Mark sees #3 and #4 as related. And this is something I do not do much of in my teaching. I do give SBAR grades for good progress on hard problems, instead of credit for right answers. I propose extensions for writing, and have optional assignments that can be very challenging. Is that enough?

I think near-impossible is affecting me as a mathy type. The idea of a challenge, that a learner would remember solving or trying is probably the goal. How do we support them to give these a try, though? Much like #3, I think sharing student work on such things is probably a key part.

At Twitter Math Camp, Sasha Fradkin had a session on impossible problems. She didn't mean this kind of impossible, but I think by coincidence, it might fit the category. Something like: using three straight cuts, divide a circle up into 3, 4, 5, 6, 7, and 8 pieces. (Not all of those are possible.) One of her takeaways was to consider what do we want the learners to mean when they say 'this is impossible.'

#5 – Create alternate ways to win

In a game, of course, you're trying to win. If there is only one way to win, the game becomes boring and narrow quickly. If a multitude of strategies is available, the game is richer as people pursue different resources and strategies.

In class, this feels to me like assessment. The goal is demonstrated understanding. If the only way to do that is timed tests, I think that narrows the game. Now it's not competitive, maybe, and the people who are better at that don't necessarily bar others from success... unless it creeps into your test writing. Or you curve. Or you measure the middle and less successful students by those who are good test takers.

For me in college there was a strange thing. My first two years I was adjusting from high school's low expectation tests to honors courses where they wanted some version of deep understanding. I got some Bs. The high school tests just wanted recall, which due to no credit of my own was easy. I couldn't not know a lot of those things. But then, beginning of my junior year, tests just made sense. I wasn't any better of a student, but I think I went almost two years without missing a question. It was weird.  When I started teaching, this got me to include a lot about test taking strategies in my classes and review days.

Eventually, though, I realized that this meant that tests weren't doing what I wanted them to do. So now my learners know the standards they're being assessed on and there are multiple ways to demonstrate understanding. And they can reassess.

#6 – Allow players opportunities to interact with other people where the outcome is based on the interaction

I think this is a regular feature of classes that feature cooperative learning.  It does require communication that is not teacher <-> student. If your classroom communication is you talking or asking questions and people answering you or asking you questions, it is one dimensional in a three dimensional world.

#7 – Give players the ability to customize, allowing them opportunities for creativity

This is sooo hard. But, ultimately, necessary. Dave Coffey likes to say that if the only choice students have is to do something or not to do it, of course some will choose not to do it. Even if the choice is as simple as choose the even or odd problems to do can increase engagement. Is it possible to let students choose a topic? Form of an assessment? Application? Which question to investigate in a 3-Act?

I love Elizabeth Statmore's emphasis on returning authority to the learners. This is part of that. Give choices and ask them why they chose as they did. Math class does not have to be everyone doing the same thing at the same time. Choices imply there is self-assessment to do. To me, this is the holy grail of assessment: learners start to think for themselves about what do they understand and what do they not get yet. And what should they do about it.

Sometimes I describe Magic as chess where you get to build your own pieces and bring your half of the board. (Plus a layer of variability from being a card game.)

Endgame

Mark's last words:

Narrative equity isn't a lens you have to view every game component through, but it is something you should view some of them through. When putting your game together, be aware that you have a lot of control over what the end experience will be. By making certain choices, you can maximize those choices that lead to your players forming stories, which in turn will change how your players emotionally bind with your game.

I am left with questions. What stories will my learners tell about the course they had with me? Will they be the hero or at least the protagonist in those stories? Will it change their view of the mathematics genre? Will every learner get an opportunity to weave a tale?

PS: Flavor Flav

I must have three unfinished blogposts to get through, but this is what I keep coming back to this week.

Natasha Lewis Harrington is a doctoral psychology student who writes about my favorite game (Magic: the Gathering) in her spare time. Sometimes she crosses the stream to great effect. Like this week, when she wrote about why this game is so good at encouraging creativity among players. It's applying the work of (let me copy and paste here) Mihaly Csikzentmihalyi (specifically Creativity: Flow and the Psychology of Discovery and Invention [Google book preview]) to the question of how can we learn to engage more. I think it's well readable by non-Magic players, so please do peruse.

Here's the quick take:
(The little bit of art is from Flickr, Paolo Colacino who does what he calls generative art. Quite neat.)

Csikzentmihalyi has a TED talk about leaving boredom:


Why is this gripping me so? Because of the divide between math as taught and math as it could be.

Math, as it is often taught, violates all three of these principles. (1) We tell you the problems to do, (3) we insist on solo mastery and uniformity of method.

Wait, that's only two.

I'm wondering if I have, in my need to change (1) and (3), more than occasionally neglected (2). Is that the procedural knowledge which I de-emphasize?  I usually do that in an attempt to get the pendulum swinging in the other direction, but in doing so am I denying needed support?

Maybe not. Maybe Learning the System in mathematics is not the procedural stuff. Maybe it's the processes, hidden behind the procedural emphasis. (The processes now appearing with their new band, the Standards for Mathematical Practice.)

Of course, there's hope. Teachers like Fawn Nguyen, Michael Pershan and Andrew Stadel are knocking this engagement issue out of the park on all three principles.

But, as Dave Coffey has cautioned, and convinced me, we need to teach our students to take control of their own engagement. So when they leave Jim Pai's classroom, they can be engaged the next year, too.

That's empowerment, and that's what I want for my students.

Trying to document our senior student teacher seminar. Lesson by Dave Coffey, (@delta_dc, Deltascape)

Dave’s quote for the day:

You must be the change you wish to see in the world. – Mohandas Gandhi

We watched part of Alan November’s TEDxNYED talk (picking up after the barbershop, about 6 min in):






What do the novice teachers notice?

Overheard snippets:
I’ll do extra work all the time. Write a letter…
Empowering students…
Present the idea of what they are learning…
Could give them a topic, send them to research, and what do you get.

Whole class discussion:
A student shares: I’m at a progressive school, and it’s hard to think about going back. The book isn’t example, example, exercise. They’re an idea, a goal, an objective, and then it’s an investigation. The problems guide you through finding the information about the topic. The parent would have to do the whole investigation to help. I did a demonstration of communicating what you’re doing, and then the students were responsible for being able to do that. Putting the work on the students is what we’re doing, and we’re there to guide them.  On the test had a check question and 19/20 had the quadratic formula right.

Q:  What’s a way for this to work in another classroom?
Dave wonders about:

Platform	Audience	Purpose
class blogs	Parents	here’s what we’ve been doing
class blogs	Peers	here’s what you missed
letter	...	...

Student response

• Home or class work? Up to you.
• Thinking of students that don’t have access to a computer… we had a designated note-taker. If someone missed they could copy.
• Saw a student teacher have learners come up to be a scribe.

Not about can do or can’t do. About can do and not yet.

One student opines:  I wish they’d say this in their videos. They come off as you have to change everything. He mentioned that students say they’ll do things for their fiends, and I will, too. But is that what you want from me as a student? But if it was just me, I wouldn’t do this portfolio. (No offense.) To totally allow students to do what they please. “It’s so nice, it’s a great metaphor…” But in reality it’s crazy. Are you kidding?

The teacher of the JK Rowling fanwriter said – she’s not a good student. Could the teacher meet her halfway? There are probably things desired for her that are not met by writing like Harry Potter.

Teachers are more important than ever, to provide that structure.

by Priki @ Flikr

A student shares:  I covered triangles and they constructed definitions, and classifications. Homework was a brochure or a puzzle. Had to have name and properties and a picture for each kind. Non-homework doers was cut in half. Next day quadrilaterals. They could make a bumper sticker or a questionairre for interviewing a quadrilateral. Ask questions, but you can’t ask the type. Again the majority of kids turned that in.

What is working? What isn’t working? What will increase engagement? This example had choice, a framework, a purpose…

We then offered a choice to work on portfolios, presentations, or cooperatively planning engaging lessons.  No one chose the lessons, with so much hanging over there heads.  (A mini-lesson for us.)  Still, it was a lively discussion and helped us process Dr. November's TED talk.

(This is me blogging Dave's lesson.  I like taking notes - probably the thing about class I miss most.)

We started class thinking about our six word teaching philosophies. How these can be the core of our planning, instruction and how we interact with students. Dave's - "Engagement that fosters capacity and agency."

From student blogs during the week, we know that the idea of evaluation was still in process for many of the student teachers.

Evaluation

• What can we do?
• What are we trying to do?
• What comes next?

It's natural to start with ourselves, that's what we know best.  And we may be saddled with assessment data that's not accountable nor reliable.  Eg. Star Math (as a TA pointed out) or MEAP data (Michigan's state Grade 3-8 assessments).

Dave shared his early classroom management by carrot and stick, and mostly sticks; he relates that to an instinct to control.  There's a time for that, but it's not always.  Then the cajoling.  There's a place for that, but not always.  Want to get to a classroom where it's about choice.  Students choose to be a part of class.  It's not easy.

Cambourne's Conditions of Learning.  ("Toward an educationally relevant theory of literacy learning:  Twenty years of inquiry," Brian Cambourne, The Reading Teacher, 49(3), 182-190.) (paraphrased)

Engagement occurs when learners are convinced that:

1. They are potential doers of these demonstrations they are observing.
2. Engaging with these demonstrations will further their purposes for their lives.
3. They can engage and try without fear of physical or psychological hurt if their attempts are not correct.

Students were asked to develop their rubrics on  Classroom Management:

:-)	:-|	:-(
Students do self-discovery activities. Teacher uses 10 second rule. Be able to get all students engaged. Understanding directions and interested because they see importance. The students are talking to other students about the lesson and asking questions. Creates a safe/welcoming environment and fosters an "I want to try," or "I think I can" atmosphere. Students excited to learn on their own.	Mostly teacher-centered. Ignoring actions & sometimes loses cool. Some students are engaged in lesson. Understanding, but no interest while doing work. They seem interested in what you are doing, but don't understand the lesson. Provides a variety of activities and gives learners a choice. Doing the work or trying, but not enjoying it.	Completely teacher-centered. Teacher acts impulsively. Majority of the students (or all)  aren't engaged. Lack of understanding, confused as to what they are supposed to do. The students are not talking to one another nor asking questions about the lesson. Has a controlling environments. Students not even trying.

He shared the post from miss brave, a 3rd grade teacher in NYC, on being disengaged.  Engagement as it relates to classroom management.

Finally, he demonstrated what his lesson planning was like in different stages of his career.  (Hopefully we'll have video of this, also.)  We can use the rubric as a landscape of progression.  What does it mean to plan? What am I focusing on at each stage?

:-)	"The learner will..." objectives!  Letting learners take control of their learning.  Tied everything to content, but also what you wanted your students to look at beyond that.  Evaluation process with students: can, trying, next  Better understanding of these particular learners, not just prior students.
:-|	Looked at what prior knowledge students might have and how it related. (Launch)  Started to put more emphasis on reasoning and justification - process in general.  More thinking about how students would respond. Changing questions to better suit students.  Making changes based on what happened last time.  Used words like 'construct' and 'consolidate.' More comfortable with some educational theory.  More of probing for understanding. More assessment.
:-(	Focusing on yourself and what you were doing. Teacher centered.  No objectives.  Close to what the book had in place.  There wasn't much wiggle room for how the lesson could be individualized, or varied depending on how the lesson goes.  Lesson plan is vague. A substitute would have no idea what to expect in terms of student difficulties.

We have to start by beginning with ourselves.  Those are bad words - teacher centered - but it's where he had to start.  "Constructivism gone mad," when he tried to jump right to student centered.  I go through these stages more quickly, but still go through them.

A student pointed out that there should also be progress throughout the year. Yes!  Gradual release... but that's for another day.

Dan Meyer's excellent TEDx talk on Math Curriculum (and problem solving, and tech enhancement...):


What do you care enough about to make the subject of your 10-20 min TEDx talk?  You can find Mr. Meyer's excellent blog at http://blog.mrmeyer.com/, called dy/dan.  I saw this video first via Michael Paul Goldenberg. 

There's a kind of interesting web of blog posts on what interest is and how to generate it.  I'd start at wehrintheworld, which I found via Ben Casochna. (an entrepeneurial site.  Hmm.)  I'm interested because of how interest relates to engagement.  Which is Holy Grailish for me.

On an almost total aside, I've also been deeply interested in the fantastic images from Iceland's volcano, from two different educational sites:

• http://www.universetoday.com/2010/04/18/incredible-images-of-iceland-volcano-from-just-a-few-kilometers-away/
• http://www.swisseduc.ch/stromboli/perm/iceland/eyafallajokull_20100414-en.html 

 Both came up in What's Hot in Google Reader, so hat tip: internet.