dy/dan

It's great, first of all, that Khan Academy has all their student exercise code on GitHub for everybody to see. I don't know any other adaptive system that does that. I figured there had to be a better way to reward them for that transparency than the criticism and judgment I'm about to post here, so I made them a badge also.

Their code illustrates the different ways good math teachers and good programmers try to figure out what students know.

Take proportions, for instance. Here is the code that runs beneath Khan Academy's proportions assessment.

In each of the dozen files I've reviewed, Khan Academy first generates some random numbers that meet certain criteria. In the proportions assessment, they call for three random unique integers between 5 and 12. No decimals. No negatives. No zeroes.

var numbers = randRangeUnique( 5, 12, 3 );

Then they use those numbers to generate exercises. With proportions, they insert an "x" randomly into that list of numbers. The final order of that list determines the proportional relationship that students will have to solve.

numbers.splice( randRange( 0, 3 ), 0, "x" );

But good teachers are more than random number generators. They create exercise sets that increase in difficulty, that ask students to demonstrate mastery in different contexts, all because proportions are conceptually difficult but procedurally simple. It's extremely easy for students to get by on an instrumental understanding of proportions alone. (eg. "All you hafta do is multiply the two numbers that are across from each other and divide by the number across from the x." Boom. It's badge time.) It's especially easy when the only thing that changes about the problem is the random numbers.

But forget good teachers for a minute. Let's look at the bar set by various standard-setting organizations. Here is what you have to do to demonstrate mastery of proportions on a) Khan Academy, b) the California Standards Test, c) the Smarter Balanced Assessment.

Khan Academy

You'll do a handful of problems just like this, with different random numbers in different places.

California Standards Test

Smarter Balanced Assessment

The difficulty and value of the assessments clearly increases from Khan Academy to the CST and then Smarter Balanced. (I'm hesitant to guess how well a student's score on the Smarter Balanced Assessment will correlate to all her practice on Khan Academy.)

Here we find a difference between good math teachers and good programmers. The good math teacher can create good math assessments. The good programmer can make things scale globally across the Internet. The two of them seem like a match made in math heaven. Just get them in a room together, right? But the very technology that lets Khan Academy assess hundreds of concepts at global scale — random number generators, string splices, and algorithmically generated hints — has downgraded, perhaps unavoidably, what it means to know math.

2012 Dec 13. Peter Collingridge points out in the comments that Khan Academy has a proportions assessment comparable to the California Standards Test. If they have anything similar to the Smarter Balanced Assessment, please let me know.

Sadie Estrella's class worked on Pixel Pattern and then watched the answer.

I'll try not to be ideological here about photos and videos in modeling tasks. If you have another way to achieve the same cathartic reaction we find 38 seconds into the video, drop me a note in the comments. I'll take it.

Previously

• You Don't Have To Be The Answer Key
• It's Killing Me. I Gotta Know.

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.

The Center for the Study of Mathematics Curriculum invited me to give a talk last week on digital math curricula. I described how print curricula limit the experiences we can offer our math students and then I made five recommendations for designing experiences digitally:

1. Show, don't tell.
2. Introduce the task as early and concisely as possible.
3. Climb the entire ladder of abstraction.
4. Crowdsource patterns.
5. Prove math works.

Any questions or criticism, please don't hold back in the comments. I also have limited availability for consultation on these kinds of projects. Drop me a line at dan@mrmeyer.com.

2012 May 1. Here's the feedback [pdf] from the academics at the conference.