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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

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

Frank Noschese, on last week's ceiling fan:

I'm dying to see the third act.

Ginny, a participant in my qualifying study at Stanford, on the water tank:

I'm dying of curiosity. Is that anywhere near the right answer?

Andrew Stadel's student, on the path of the basketball:

Can we watch the video to see if he makes it? It's killing me. I gotta know.

All three describe the experience of not knowing the answer to a math problem as something like death. A math problem. How does that happen?

My best guess? You start with a credible document of the world your students live in. That could be an actual water tank in the classroom or a representation of a water tank on video. It has to be credible. Then you document something happening — the tank filling, the fan spinning down, the ball sailing through the air — long enough for a learner to have a sense of what is happening and what might happen next.

That's where you end the document. Then work happens. The work is motivated in part by the student's knowledge that the answer actually exists, that the teacher talks a huge game about math being everywhere and in everything and we're about to put that to a test.

Then you show the answer.

Please watch this video of Ginny watching the answer to the water tank problem. This moment was incidental to my actual research question. I have no way of knowing if Ginny would have experienced the same mixture of suspense, elation, and catharsis reading the answer to the same problem in the back of her textbook. I only know that if you had told me in my first year teaching that suspense, elation, and catharsis were possible reactions to a math problem, as much as I loved math myself, I would have thought you were crazy.

Previously: You Don't Have To Be The Answer Key, Handle With Care.

2012 Oct 2. Rachel Kernodle writes about Bean Counting: "… the 4 groups that correctly got the extension with no help from me literally SCREAMED and high-fived each other when I played the answer video …."

2012 Oct 2. Chris Robinson writes about Taco Cart: "More student comments from @ddmeyer 's Taco Cart #3act: I'm losing sleep over the answer, this problem is killing me. Teachers, #3act works. Students made me replay the answer to @ddmeyer's Taco Cart #3act so they could provide play-by-play in the style of a horse racing announcer."

Apple's count-up to 25 billion apps struck the same chord for several of us last week. Three of us tried to represent that moment for students with photos and video. I don't find the question, "Who did a better or worse job?" as interesting as "What were the principles that organized each of our work?"



Dan Anderson


Sean Dardiss


Sean has a timer rolling in the foreground. Dan has superimposed his computer's clock over the counter. Those design choices interest me. When he first saw the counter, I'm wagering Dan didn't have his clock open. I'm positive Sean added that timer later in AfterEffects. What were they trying to accomplish with those additions?

A guess? They were trying to make the problem solvable, which is probably the most natural inclination for a math teacher designing a task for a math class.

I want to be explicit about my M.O. here without calling it better or worse than theirs. I'm still trying to figure this out.

  1. I want to recreate as exactly as possible the moment when Apple's counter perplexed me, when it dialed the pressure in my head up to eleven and made the question irresistible, "When should I start bombarding the app store with downloads if I want to win $10,000?"
  2. I want to separate the tools, information, and resources I used to answer that question from the perplexing moment itself.

The first point argues for a recording of the screen just the way it was when I first found the counter. Nothing extra.

The second point argues for postponing — not eliminating — a) data samples, b) a table, c) graphing paper, d) the slope formula, e) a lecture about the point-slope form of a line, etc., until after we've settled on a perplexing question that needs those tools, resources, and information.

This is a more accurate representation of how I solved the problem. (I had to decide that a table and a graph would be helpful. I had to decide that a linear equation would be the best model. No one gave me any of that.) It's also more perplexing to see a problem as it exists in the wild, "posed simply and innocently, not flayed alive by terminology, labels, and notation."

One More Example

I've created two visual prompts for the same question, "How many gumballs are in the machine?" (This picture is from Dan Anderson, also.) One version abstracts the problem at the same time that it tries to perplex students. The other postpones that abstraction for just one moment. Both will result in (more or less) the same mathematical analysis. I'm curious which one would perplex students more.

It would be nice to have a website to test out the difference a little more empirically.

Featured Comment


I had kids staring at the screen with their iPhones out waiting to download at the beginning of class. It was hilarious. One of the most engaging problems we’ve done all year!

The medium is the message. The medium defines, changes, and distorts the message. The words "I love you" mean one hundred different things spoken by one hundred different people. Those words convey different meanings spoken on the phone, written on a fogged-over bathroom mirror, and whispered bedside in a hospital.

YouTube videos, digital photos, MP3s, PDFs, blog posts, spoken words, and printed text are all different media and they are all suited for different messages. When you attempt to distribute mathematics through any of these media, it changes the definition of mathematics.

Silicon Valley's entrepreneurs, venture capitalists, and big thinkers assume a shared definition of "mathematics." They innovate around the delivery of that mathematics. CK-12 has PDFs. Khan Academy has YouTube videos. Apple has iPad apps. ALEKS and Junyo have computer adaptive tests. Very few of them understand that each of those delivery media changes the definition of mathematics.

Even worse, at this moment in history, computers are not a natural working medium for mathematics.

For instance: think of a fraction in your head.

Say it out loud. That's simple.

Write it on paper. Still simple.

Now communicate that fraction so a computer can understand and grade it. Click open the tools palette. Click the fraction button. Click in the numerator. Press the "4″ key. Click in the denominator. Press the "9″ key.

That's bad, but if you aren't convinced the difference is important, try to communicate the square root of that fraction. If it were this hard to post a tweet or update your status, Twitter and Facebook would be empty office space on Folsom Street and Page Mill Road.

It gets worse when you ask students to do anything meaningful with fractions. Like: "Explain whether 4/3 or 3/4 is closer to 1, and how you know."

It's simple enough to write down an explanation. It's also simple to speak that explanation out loud so that somebody can assess its meaning. In 2012, it is impossible for a computer to assess that argument at anywhere near the same level of meaning. Those meaningful problems are then defined out of "mathematics."

Do you want to know where this post became useless to Silicon Valley's entrepreneurs, venture capitalists, and big thinkers? Right where I said, "Computers are not a natural working medium for mathematics." They understand computers and they understand how to turn computers into money so they are understandably interested in problems whose solutions require computers. Sometimes a problem comes along that doesn't naturally require computers. Like mathematics. They may then define, change, and distort the definition of the problem until it does require computers.

Some companies pretend those different definitions don't exist. They pretend that we all mean the same thing when we talk about "mathematics." Khan Academy acknowledges the difference, though, and attempts to split it by saying, in effect, "We'll handle the math that plays to our medium's strength. Teachers can handle the other math." So Khan lectures about things that are easy to lecture about with computers and his platform assesses procedures that are easy to assess with computers. Teachers are told to handle the things for which teachers are a good medium: conversation, dialogue, reasoning, and open questions.

That delegation only works to the extent that teachers and computers convey complementary definitions of mathematics. But the message from Silicon Valley and the message from our best math classrooms contradict one another more often than they agree. On the one hand, Silicon Valley tells students, "Math is a series of simple, machine-readable tasks you watch someone else explain and then perform yourself." Our best classrooms tell students, "Math is something that requires the best of your senses and reasoning, something that requires you to make meaning of tasks that aren't always clearly defined, something that can make sense whether or not anyone is there to explain it to you."

I won't waste any effort complaining that my preferred definition of mathematics has been marginalized. That effort can be better spent. Anyway, in every way that affects Silicon Valley's bottom line, the Common Core State Standards have settled that debate. Mathematics, as defined by the CCSS, isn't just a series of discrete content standards. It contains practice standards, too: modeling, critiquing arguments, using tools strategically, reasoning abstractly, and others. The work of mathematicians. Any medium that tries to delegate one set of standards to computers and the other to teachers should prepare for a migraine.

Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.

Has your ed-tech startup been struggling to demonstrate statistically significant gains on the California Standards Test, which features tasks like this:

That's your home-turf. Simple, machine-readable assessments. It will never get any easier for you than that. How much worse will your results look when we assess the same standard in 2014 with tasks that connect mathematical content to mathematical practices:

The medium is the mathematics. How does your medium define mathematics and is that definition anything that will be worth talking about in two years?

Full Disclosure: I'm a doctoral student at Stanford University in math education. I was a high school math teacher. I consult with ed-tech startups as time allows. I also develop digital math curricula that I sell to publishers and give away online.

Comment Policy: My usual policy is to close comments on posts that mention Khan Academy because they get silly almost instantly. But Khan Academy is only a symptom of a sickness that's gripped this valley for as long as I've lived here. That sickness interests me and your thoughts on that sickness interest me. I'm leaving comments open but I'll trashcan anything that doesn't enhance our understanding of that sickness. That includes "Attaboys," etc.

2011 Feb 7. Neeru Khosla, founder of CK-12, responds in the comments.

2011 Feb 7. Josh Giesbrecht posts a useful reply at his own blog, focusing on technology as an assessment, not technology as a medium.

2011 Feb 7. Web Equation from VisionObjects does a fantastic job translating scribbles on the screen to LaTeX.

2011 Feb 7. Silicon Valley's (unofficial) rebuttal to my post is at Hacker News. Let me excerpt a few responses.

jfarmer poses a very productive alternative to my thesis, "What is technology good for?" rather than "How does technology change mathematics?":

In design, a skeuomorph is a derivative object that retains some feature of the original object which is no longer necessary. For example, iCal in OS X Lion looks like a physical calendar, even though there's no reason for a digital calendar to look (or behave) like a physical calendar. The same goes for the address book.

This is what I see happening in online education. I don't think it's a case of "lol, Silicon Valley only trusts computers," but rather starting off by doing the most literal thing.

Textbooks? Let's publish some PDFs online. Lectures? Let's publish videos online. Homework and tests? Let's make a website that works like a multiple-choice or fill-in-the-blank test.

These are skeumorphs. There's no reason for the online equivalent of a textbook to be a PDF, it's just the most obvious thing.

For me it's 1000x more interesting to ask "On the web, what's the best way to do what a lecture does offline?" than to say "Khan Academy videos are the wrong way of doing it."

Arun2009 offers a common view, that mathematics is many different things to many different people:

The trouble with trying to arrive at any single definition of Mathematics is that Mathematics is different things to different people. A research level Mathematician might see it differently (finding patterns, abstraction, theory – axioms and proofs) from an Engineer who has a purely practical interest in it (cookie cutter methods and formulas). For everyday use Mathematics is a set of algorithms for doing stuff with percentages, fractions, basic arithmetic etc.

He's absolutely right, but if we're pragmatic in the least, we'll ask "which of those definitions does the most good for students?" and we'll look at the Common Core State Standards, which is the de facto definition of mathematics for those students. (My appeal to the CCSS was an attempt to reduce exactly this subjectivity.)

Various hackers took me to task for claiming it's difficult to represent mathematical notation using computers. ie. the square root of 9/4.


Square root ? "(3/4)^(1/2)" or maybe "sqrt(3/4)". There's no complexity in parsing that. I do agree it is not as natural as on paper but maybe tablets will find a way to improve that. Thats what innovation is here for after all.

japhyr rebuts convincingly, in my opinion:

He is writing about math education. The characters (3/4)^(1/2) make sense to all of us who have already learned math and know some programming languages, but that syntax is pretty confusing to students who are just developing a real understanding of exponents.


One could make the argument that any mathematical syntax is equally confusing for the novice–so why not start them on something they'll be using later anyways?

That's an interesting idea but unless we're also positing a universe with utterly ubiquitous computing, we'd be better off preparing students to communicate in media that are readily available. What if a computer isn't available for our students to code some LaTeX to express themselves?

davidwees illustrates my overall point well. Does this seem natural to anybody?

\png \definecolor{blueblack}{RGB}{0,0,135} \color{blueblack} \begin{picture}(4,1.75) \thicklines \put(2,0.01){\arc{3}{3.53588}{5.8888}} \put(.375,.575){\line(1,0){3.25}} \put(1.22,1.375){\makebox(0,0){\footnotesize$ds$}} \put(.6,.5){\makebox(0,0){\footnotesize$x=0$}} \put(3.36,.5){\makebox(0,0){\footnotesize$x=\ell$}} \dottedline{.05}(1.0,.575)(1.0,1.10) \put(1.0,.5){\makebox(0,0){\footnotesize$x$}} \dottedline{.05}(1.5,.575)(1.5,1.40) \put(1.5,.5){\makebox(0,0){\footnotesize$x+dx$}} \put(1.22,.65){\makebox(0,0){\footnotesize$dx$}} \dottedline{.04}(0.6,1.12)(1.25,1.12) \put(1.0,1.14){\vector(-1,-1){.45}} \put(.58,0.83){\makebox(0,0){\footnotesize$T$}} \put(.77,1.05){\makebox(0,0){\scriptsize$\theta(x)$}} \put(1.18,1.16){\makebox(0,0){\scriptsize$\theta(x)$}} \dottedline{.04}(1.5,1.41)(2.1,1.41) \put(1.5,1.44){\vector(4,1){.67}} \put(2.22,1.59){\makebox(0,0){\footnotesize$T$}} \put(1.95,1.45){\makebox(0,0){\scriptsize$\theta(x+dx)$}} \end{picture}

Symmetry, takes the conversation to Khan Academy:

As to why many people might want to defend Khan Academy, well, its because I think I would have been much happier with Khan Academy than the math education I actually had, and I would very much like it to be available to children like myself. I was bored stiff in math class in middle and high school, and being able to work at my own base, not bound by the slowest person in the class, would have been amazing.

This same sentiment crops up in the comments thread here and I think it's utterly on point. Teachers are a great medium for lots of things that a YouTube video isn't. "Conversation, dialogue, reasoning, and open questions," as I put it in my post. If you, as a teacher, aren't taking advantage of your medium, if you're functionally equivalent to a YouTube video, you should be replaced by a YouTube video.

Sudarshan summarizes that elegantly:

Incompetent/bored math teacher < khan academy < better online learning platform < Good math teacher.

FWIW, I stopped by that thread and summarized my argument in three lines:

  1. There are different ways of defining mathematics, and some of them contradict each other.
  2. Silicon Valley companies wrongly assume their platforms are agnostic on those definitions.
  3. For better or worse, if you're trying to make money in math education, the Common Core State Standards are the definition that trumps your or my preference for recursion, computational algebra, etc, and those standards include a lot of practices for which, at this point in history, computers aren't just unhelpful, but also counterproductive.

2011 Feb 10. Zack Miller e-mailed. Zack graduated from Stanford's teacher training program. He now teaches at the charter school where I've been consulting with my adviser and two other graduate students. He's a proponent of blended learning, individualized instruction, and Khan Academy, in particular. He gave me permission to excerpt his e-mail. I'm going to post the entire thing, bold one line, and make a comment on that line at the end.

Students’ mathematical learning and sense-making has a much higher ceiling if the material they encounter follows a coherent mathematical narrative that is relevant to and keeps pace with their current individual mathematical understandings.

The above is not a reality in most math classrooms. Students’ math understandings are Swiss cheese, largely because time, not competency, was fixed in their math education. What’s standing in the way of the above becoming a reality? Resources and good data. 1 teacher must provide 100 unique learners with learning experiences, daily. Can our country’s median teacher (or best teacher) provide a learning experience that is in each student’s zone of proximal development every day (and also be an expert in classroom management, curriculum, parent communication, data analysis, etc.)? No. So we’ve looked for ways over the years to ameliorate the situation (tracking, heterogenous classes, differentiation, etc.).

Silicon Valley is trying to innovate because students’ mathematical learning has a much higher ceiling if curriculum always meets their ZPD, and perhaps now there are ways we can offer that. THIS IS BLENDED (not to be confused with computers replacing traditional classrooms with traditional video lectures). Yes, the medium matters – without question – but breaking from the ratio of “30 students:1 pace” matters more. Any tool that gathers relevant data and/or provides good learning experiences helps me individualize, and (like many of your commenters) I disagree that there is no computer-based tool that does this (Geogebra, for one, was great for today’s lesson that you and your team designed!).

I agree that, at present, Silicon Valley’s current tools are limited. Pedagogy is weak. But I disagree that doing this:

"We'll handle the math that plays to our medium's strength. Teachers can handle the other math."

Implies also doing this:

Silicon Valley tells students, "Math is a series of simple, machine-readable tasks you watch someone else explain and then perform yourself."

Isn’t it possible to frame computer-based tools in a way that supports your definition of math (a definition that I happen to closely align with)? Doesn’t procedural fluency (which will still be a large portion of Common Core assessments, I believe) support reasoning, problem solving, and sense-making?

And lucky for our students, people like you will help Silicon Valley with their pedagogical shortcomings and misguided definitions of math. As the quality and quantity of “learning experiences offered” improve, students can have a more individualized experience, and that high ceiling can be realized.

My point is that the tools that allow you to individualize math instruction are not neutral on the question, "What is mathematics?" My second point is that not every answer to the question, "What is mathematics?" harmonizes with every other answer. Sometimes they clang around against each other awkwardly.

2011 Feb 15: EdSurge offers a summary of this post and its responses over at Fast Company.

2011 Feb 15: Just noticed (way late) that Jacob Klein, one of the founders of Motion Math, responded here. Klein is hip deep in the culture I'm describing here and he makes games that help students practice mathematical skills I value, so give him a look.

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