Month: October 2012

Total 18 Posts

What Do Adaptive Math Systems Really Know About What You Know?

First, Michael Goldstein:

Khan Academy alone gives the following information: time spent per day on each skill, total time spent each day, time spent on each video, time spent on each practice module, level of mastery of each skill, which ‘badges’ have been earned, a graph of skills completed over number of days working on the site, and a graphic showing the total percentage of time spent by video and by skill.

Second, Jose Ferreira, CEO of Knewton:

So Knewton and any platform built on Knewton can figure out things like, “You learn math best in the morning between 8:32 and 9:14 AM. You learn science best in 40-minute bite-sizes. At the 42-minute mark your clickrate always begins to decline. We should pull that and move you to something else to keep you engaged. That thirty-five minute burst you do at lunch every day? You’re not retaining any of that. Just hang out with your friends and do that stuff in the afternoon instead when you learn better.”

I don’t have a lot of hope for a system that sees learning largely as a function of time or time of day, rather than as a function of good instruction and rich tasks. It isn’t useless. But it’s the wrong diagnosis. For instance, if a student’s clickrate on multiple-choice items declines at 9:14 AM, one option is to tell her to click multiple-choice items later. Another is to give her more to do than click multiple-choice items.

These systems report so much time data because time is easy for them to measure. But what’s easy to measure and what’s useful to a learner aren’t necessarily the same thing. What the learner would really like to know is, “What do I know and what don’t I know about what I’m trying to learn here?” And adaptive math systems have contributed very little to our understanding of that question.

For example, a student solves “x/2 + x/6 = 2” and answers “48,” incorrectly. How does your system help that student, apart from a) recommending another time in the day for her to do the same question or b) recommending a lecture video for her to watch, pause, and rewind?

Meanwhile, these trained meatsacks have accurately diagnosed the student’s misunderstanding and proposed specific follow-ups. That’s the kind of adaptive learning that interests me most.

Featured Comments

Chris Lusto:

But then we’d need like an entire army of trained meatsticks, each assigned to a manageably small group of students, possibly even personally invested in their success, with real-time access to their brains and associated thoughts, perhaps with a bank of research-based strategies to help guide those students toward a deeper understanding of…something.

That seems an awful lot like a world without clickrates, and I’m not sure it’s a world I want to live in. Or maybe I’m just cynical between 11:30 and 12:00, on average, and should think about it later.

Dan Anderson:

A big advantage with meatsacks over computers is the ability of a human to look at the work. Computers can only indirectly evaluate where the student went wrong; they can only look at the shadow on the ground to tell where the flyball is going. Meatsacks can evaluate directly where the student is going awry.

1,400 Rectangles

Some math teachers were sharing dinner following last week’s Northwest Math Conference when Marc Garneau said something truly implausible:

If you have a class of students draw a rectangle, they’ll combine to create the golden rectangle.

Truly implausible, but Marc stood by it, along with at least one other member of our party. Dave Major set up a web page so we could collect data. You all obliged us with 1,400 rectangles, about a third of which I’ll show you in this video:

Mean: 6.16; Median: 2.087; Standard Deviation: 18.296. So, no, not the golden rectangle. And now Marc owes me a new car.

a different dave wrote:

I predict that the shape of the rectangles is going to be very heavily influenced by the shape of the canvas provided.

Not that either. Now a different dave owes me a new car too.

Here’s all the data. Tell us something interesting about them we don’t already know.

The Necessity Principle

How could we improve this task?

Fuller, Rabin, and Harel (2011) [pdf] define “intellectual need,” “problem-free activity,” and offer several ways to improve that task in one of the best pieces I read last summer:

When students participate in mathematical activities that stimulate intellectual need, we say that they are engaged in problem-laden activity. Unfortunately, many students are engaged in problem-free activity, in which they are driven by factors other than intellectual need and, as a result, do not have a clear mental image of the problem that is being solved, or indeed an understanding that any intellectual problem is being solved.

The piece features:

  • Dialog between teachers and their students that results in “problem-free behavior” and “social need.” There’s something in here for everybody. Everybody – myself included – will feel a twinge of recognition reading one or more of those exchanges.
  • Great suggestions for how to mend those scenarios, for queueing up intellectual need and problem-laden behavior.
  • Five categories of intellectual need. The need for certainty, causality, computation, communication, and connection. You can lean on any of those categories and watch several great lesson ideas fall out.

Featured Comment

mr bombastic:

The recursive part in the original question is especially annoying in that it sends the message that math is used to take something that is totally obvious (two more brick in the next row) and somehow make it seem complicated.

Dave Major Shows You The Future Of Math Textbooks

I’ve been trading e-mails over the last few weeks with Dave Major, a teacher in Dubai who also knows how to use code to make dreams come true.

For instance, I wrote a mushy love ode to the Taco Cart task of my dreams. Dave Major made it real.

Then I asked him to create an activity I described in this talk at 28:01. We ask students to create a triangle with certain specifications. They submit their triangle and then they see quickly and easily whether or not everyone else created the same triangle from the same specs. If they did, we should prove that it’s impossible to create another triangle. If they didn’t, then we have a counterexample and we can axe the hypothesis.

Dave put it together. You should check it out. He’s giving you a look at the math textbook of the future, several years early.

Featured Comment


I keep thinking of learning a programming language, but didn’t quite have a reason why. I think I have one now.