dy/dan

If you and I have had a conversation about math education in the last month, it's likely I've taken you by the collar, stared straight at you, and said, "Can I tell you about the math lesson that has me most excited right now?"

There was probably some spittle involved.

Evan Weinberg posted "(Students) Thinking Like Computer Scientists" a month ago and the lesson idea haunted me since. It realizes the promise of digital, networked math curricula as well as anything else I can point to. If math textbooks have a digital future, you're looking at a piece of it in Evan's post.

Evan's idea basically demanded a full-scale Internetization so I spent the next month conspiring with Evan and Dave Major to put the lesson online where anybody could use it.

That's Do You Know Blue?

Five Reasons To Love This Lesson

It's so easy to start. While most modeling lessons begin by throwing information and formulas and dense blocks of text at students, Evan's task begins with the concise, enticing, intuitive question "Is this blue?" That's the power of a digital math curriculum. The abstraction can just wait a minute. We'll eventually arrive at all those equations and tables and data but we don't have to start with them.

Students embed their own data in the problem. By judging ten colors at the start of the task, students are supplying the data they'll try to model later. That's fun.

It's a bridge from math to computer science. Students get a chance to write algorithms in a language understood by both mathematicians and the computer scientists. It's analogous to the Netflix Prize for grown-up computer scientists.

It's scaffolded. I won't say we got the scaffolds exactly right, but we asked students to try two tasks in between voting on "blueness" and constructing a rule.

1. They try to create a target color from RGB values. We didn't want to assume students were all familiar with the decomposition of colors into red, green, and blue values. So we gave them something to play with.
2. They guess, based on RGB values, if a color will be blue. This was instructive for me. It was obvious to me that a big number for blue and and little numbers for red and green would result in a blue color. I learned some other, more subtle combinations on this particular scaffold.

This is the modeling cycle. Modeling is often a cycle. You take the world, turn it into math, then you check the math against the world. In that validation step, if the world disagrees with your model, you cycle back and formulate a new model.

My three-act tasks rarely invoke the cycle, in contrast to Evan's task. You model once, you see the answer, and then you discuss sources of error. But Evan's activity requires the full cycle. You submit your first rule and it matches only 40% of the test data, so you cycle back, peer harder at the data, make a sharper observation, and then try a new model.

The contest is running for another five days. The top-ranked student, Rebecca Christainsen, has a rule that correctly predicts the blueness of 2,309 out of 2,594 colors for an overall accuracy of 89%. That's awesome but not untouchable. Get on it. Get your students on it.

a/k/a Dave Major Goes Bananas

Shorter: Dave Major and I are experimenting again with what math textbooks could look like on devices that are digital and networked. Our most recent experiment is Ice Cream Stand.

Longer: Last September, Kate posted this image to Twitter attached to the tweet, "Worst geometry problem ever: can't be solved until after you solve it."

Clever bit, right? Classic Kate.

We could print that out and have students use a compass and straightedge to construct the circumcenter (the point that's equidistant from all three coffee shops). That'd be a fine summative assessment. Very "real world," etc.

But if you'd like to use Kate's tweet to motivate the need for the circumcenter, to give students a reason to care about the circumcenter, we'll need to start much lower on the ladder of abstraction. We'll need to throw out formal vocabulary and formal operations for a few minutes. We'll need to start with intuition.

So we changed the domain from coffee to ice cream. We changed the environment from a roadway (a complicated space) to a park (an open space). And we gave students a few easy choices. "Which ice cream stand would you pick, given where you're standing right now?"

Students see that they're basically painting the field one dot at a time.

So we ask them to extend that metaphor and paint the entire field so that someone else can see which stand is the closest no matter where they are in the park.

This is a task that a lot of students can complete regardless of their mathematical knowledge. It's expensive, but not impossible, to provide this task on paper. It's impossible to do on paper what comes next.

We combine the entire class' park paintings.

That's a composite from three dozen people on Twitter.

Dave and I then asked students for some preliminary thoughts about how we could calculate the right painting. But that's where we finished. The point is, students now want to know, "Who's right? Who's closest?" And what's weird is that our intuition validates the math to a degree.

That is to say, you can see areas where Twitter agreed with itself. You can see areas where Twitter disagreed with itself. When you construct the circumcenter from the perpendicular bisectors, you'll find that they overlay rather neatly on the areas of disagreement.

That's the ladder of abstraction. It isn't impossible to climb it with print-based tasks, but a digital networked device makes it a lot easier.

Open Questions

• Q: Where does this activity go next? We could add some expository text about the circumcenter. We could leave that to the teacher. We could calculate which student took the best guess in her painting of the field. A huge open question throughout these projects is, "What role does the teacher play here?"
• Q: Another huge, open question is, "What happens to the first student who runs through this activity?" Her composite painting is just her own painting. Dave and I are developing activities that exploit the network effect. They get better and more interesting when more students use them. So again: what happens to the first student through?

BTW. Dave Major wrote his own post about this project.

Featured Comments

Alexandre Muniz:

The burning question I have after looking at this is, why is the average line a bit wrong? (Especially the blue/green line.)

Evan Weinberg:

The line of uncertainty shows where the intuitive power of the brain breaks down. This is where the power of mathematical tools can step in to hone in on a more precise answer. What strikes me here is that the mathematical tools don’t do that much better of a job.

Jason Dyer:

If you allow the first student through to see the picture as it gets revised (via a reload button or some auto-update), I don’t see a terrible problem (except for the usual classroom dilemma of what you do with any student that finishes fast).

a/k/a Dave Major Rides Again

It turned out to be productive and fun arguing over who among the four contestants in this video did the best job drawing a square. Video served us well. It gave us something to look at, argue about, and abstract. But video is still a static medium in many ways. The pictures are moving but it doesn't edit well. It doesn't personalize. It doesn't reflect the learner in any way.

So Dave Majors and I partnered up again to kick around an idea of what this task would look like in code, in a web browser, and came up with better best squares.

He's written a post describing some of his technical innovations. I'm going to use this space to point out our pedagogical innovations.

1. The most obvious difference here is that instead of watching four people attempt to draw a square, you get to attempt to draw a square yourself.
   
2. That quadrilateral then follows you throughout the text. Rather than using a generic example to illustrate a mathematical concept, we use the example you created. We talk about its perimeter. We talk about its area. The diagrams in the margins change. The text in the textbook changes.
   
3. You see your classmates' quadrilaterals and make an intuitive ranking of their square-ness. When we formalize the concept of square-ness later, we'll refer back to our initial rankings. Ideally, the mathematics will validate the student's intuition and vice versa.
   
4. You can revise and skip most questions. We're deviating here from our last experiment where each question had to be completed before you could move on. In a print textbook, you can always flip forward and see what's next or move onto a new task if you don't want to complete the current one. So you can leave an answer blank. You can go back and revise your answers. The textbook doesn't judge you. It doesn't say, "You're wrong." It reports your response (or non-response) to your teacher and lets your teacher make the pedagogical judgement there.
5. The teacher's edition is so useful. I asked Dave to let me see all responses disaggregated a) by student and b) by question. I want to click on Mike's name and see all his progress throughout this unit — everything he drew, everything he wrote. Then I want to click on each question and see every response. Dave went above and beyond here. You see every student response but you also see the revision history on those responses. You can trace the student's thinking. You can also flag student responses to show the class. I'm such a fan of Dave's work here.
   
6. Don't like our definition of "best square" as being the ratio of areas? Submit your own. The system will accept your formula, send it to the teacher, and then use it to rank the entire class' quadrilaterals.
   

Dave and I both agreed this problem is a little too obscure and weird to justify all the effort we put into it. But critique the digital pedagogies rather than the task itself. These pedagogies can transfer to other, better tasks. Critique this definition of personalized learning.

Previously. Dave Major Shows You The Future Of Math Textbooks.

2013 Mar 27. A UK teenager codes the algorithm for judging the best circle. Be sure to stick around for the part where the cat judges you.

Five years ago I released a collection of 10 fifteen-second videos that helped orient my students to abstract and graphical representations.

Kids like them.

@ddmeyer Kids went nuts over GRAPHING STORIES. Gets 'em every time. Thanks again for making and sharing these!!!!!

— CheesemonkeySF (@cheesemonkeysf) December 3, 2012

Last year I asked you guys to submit your own graphs and stories which I edited together by hand.

Today, in a joint collaboration with the BuzzMath team, we're releasing 24 of those videos for immediate download and use in your classrooms, all tagged by content and math. (ie. "a step function about ponies")

You guys were way more creative than I had anticipated:

• Check out Christopher Danielson's step function about ponies.
• Adam Poetzel runs a master class on abstraction by graphing the same video two different ways.
• It's hard not to chuckle at Arianna Hoshino's students in this piecewise graph.
• Mariah Thompson's graph of Time v. Time is just bananas.

Call for Submissions (Sort Of)

I'm never gonna do what I did a year ago ever again. Editing all those videos by hand took months of my time and probably a year off my life. But I would like to know what holes you see in this library and what we can do to plug them.

Do we need more videos with periodic functions? Do we need more videos featuring bacon? Suggest them in the comments. If it's a good idea and you can film the video, I'll make your graphing story on a case-by-case basis. This thing will grow larger and awesomer.

BTW. Be sure to drop a tweet @BuzzMath thanking them for their killer work here.

We know there are important steps [pdf] you can take to ready students for an explanation of key concepts. Riley Lark is helping you do several of them very easily with his open source ActivePrompt project. While Dave Major and I continue to bat around very specific implementations of digital curricula, Riley has created an extremely open framework, useful for all kinds of purposes.

This is everything: the student sees an image and has to place a red dot somewhere on top of it according to instructions given by the teacher. It sounds too simple to be of any use.

Two Uses

Drag the red dot to where you put the cafeteria so that it's the same distance from each school.

Drag the red dot to where line m will intersect line n.

You see where this goes, right? Even with the second prompt, which isn't explicitly "real world" in the sense that we usually mean it, students now have experience with the context, which makes it real to them.

Then we start to abstract it and help students work with these concepts:

These brief experiences help immensely to set up and motivate the explanation that follows. It would be great (note to Riley) if the teacher could establish the correct answer at the end of the task (a teacher dot) which would then inform the students how close their guesses came. Also: student names on mouseover, mobile compatibility, vertical lines, and horizontal lines.

You can play with it immediately on Heroku. Be sure to link up your creations in the comments so we can all play along.

BTW. My hope in sharing Dave Major's work and Riley Lark's ActivePrompt and my own experiments is that you will become agitated and unhappy with whatever curriculum you are currently using, and that you will express that agitation and unhappiness to the people who publish and sell you that curriculum. None of us are anywhere close to nailing the question, "What do you do on day [x] with concept [y]?" for the entire set of x and y. But before we answer that question, we need to define the modern digital textbook. So here's my pullquote definition, heavily informed by Dave and Riley's work:

The modern digital textbook isn't a collection of content to be consumed. It's a collection of experiences, of which content consumption is only one part.

Riley Lark's red dot is one of those experiences.

2012 Nov 29. Riley Lark takes you behind the scenes and shows off several creative ActivePrompts.

2012 Dec 4. Learning Catalytics (a for-profit product) seems to have done a lot of good work in this area already.