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PearDeck is technology you should try out. Here’s how it works. Let’s say this image fascinates me:


It takes a certain amount of spatial skill to answer the question, “How high will the creamer be in the upside-down container? Will it be higher than the original? Lower? The same?” (I mean the volume is the same, after all.)

So I create a new PearDeck presentation and send the link out on Twitter asking just those questions. PearDeck then lets me capture the feedback of these students in realtime.


The teacher interface expands to let me know whose answers are close or not that close.


This sets me up for anything from an explanation of how to calculate solids of revolution in Calculus or a debate about covariation in Algebra.

If you’re an educational technologist and you think this is interesting, please notice that this is the opposite of individualized instruction. It’s socialized instruction. PearDeck would be much less interesting if you were the only person estimating, or if you were answering the question “Will it be higher or lower or the same?” alone.

Sometimes learning is less fun when you’re learning at your own pace.

(The answer.)

a/k/a [Makeover] Painted Cubes (See preview.)

That’s a very helpful comment from a recent workshop participant. Textbooks don’t have that same luxury.

Here’s an example. Watch how Connected Mathematics treats the classic Painted Cube problem:


Here are elements the textbook has already added:

  1. A central question. (“How many faces of the little cubes have been painted?”)
  2. A strategy. (“Look at smaller versions of the cube.” It also tells you by omission that it’s impossible to find more than three faces painted.)
  3. A table. (For organizing your data.)
  4. Table column headings. (Edge length, total cubes, total cubes of each kind.)

If you subtract those elements and add them in later, you get to ask interesting questions and host interesting conversations with your students. Like this:

  1. A central question. (“What questions do you have about Leon and his cube?” And later: “Guess how many cubes don’t have any paint on them at all?”)
  2. A strategy. (“What are all the possibilities for the number of faces that could have paint on them? Could five faces have paint on them? Can I tell you how mathematicians work on big problems? They look at smaller versions of the big problem. What would that look like here?”)
  3. A table. (“All of the numbers from our smaller versions are getting out of control. How can we organize all these loose numbers?”)
  4. Table column headings. (“What kind of data should we look at? What about these cubes seems important enough to keep track of?”)

You can always add those elements into the problem – the questions, the information, the mathematical structures, the strategy – as your students struggle and need help. But you can’t subtract them.

Once your students see the table, you can’t ask, “What tool could we use to organize ourselves?” The answer has been given. Once they see the table headings, you can’t ask, “What quantities seem important to keep track of in the table?” They know now. Once you add the strategy (“Look at smaller versions.”) their answers to the question “What strategies could we use?” won’t be as interesting.

In sum, much of the problem has been pre-formulated, which is a pity, seeing as how mathematicians and cognitive psychologists and education researchers agree that formulating the problem leads to success and interest in solving the problem.

So again I have to remind myself to be less helpful and be more thoughtful instead.

BTW. Of course I’m partial to Nicole Paris’ setup of the task:


Great Help From The Comments

I’m reprinting Bryan Meyer’s entire comment:

I don’t know that I have anything terribly insightful to add, but this seems like a fun conversation.

I don’t really see too much that is wrong with the problem/puzzle itself, which (to me) is something like:

I have this cube (show picture/tangible) made up of smaller cubes. If I dipped the whole thing in paint, how many of the smaller cubes would have paint on them? Is there a rule or shortcut we can create that would allow us to answer that question for ANY sized cube?

To me, the issue seems to be that the version we see in your blog post attempts to steer the direction of student thinking and leaves little room for play and divergent thinking/approaches. It “scaffolds” away all of the rich mathematical thinking and play in an attempt to cover standards. In particular, the unspoken assumption in the way it has been printed is that writing and graphing linear, quadratic, and exponential functions is the real “Math” in the task (things we can easily point to as belonging to the discipline/standards).

But, at it’s core and without all the mechanical scaffolding (as re-posed here), the question allows room for many mathematical strengths and habits of mind to be valued and sends different messages about what the real “math” might be: taking things apart and putting them back together, creating systems of organization, assigning variables, making generalizations, posing extension questions, etc. In addition, because it doesn’t dictate how to proceed, it encourages students to trust their own thinking and allows them to “see themselves” in the work that develops. The work of the teacher becomes to follow the student, looking for mathematically ripe opportunities in their work and thinking.

2014 Jun 2. Christopher Danielson brings his perspective to the task as writer of CMP.

2014 Jun 2. Here’s the makeover.

Painted Cubes is a classic task, canonized right alongside the Pool Border task and Barbie Bungee, but that doesn’t mean it’s beyond help, or that everyone treats it exactly the same way.

Here’s a treatment from Connected Mathematics. What would you do with this and why would you do it? (Click for larger.)


Great Classroom Action


Christina Tondevold teaches her first three-act math task. There’s a lovely and surprising result at the end, when her students realize that with modeling the calculated answer doesn’t always match the world’s answer exactly.

After they all had taken the 24 bags x 26 in each bag, every kid in that room was so confident and proud that they had gotten the answer of 624, however … the answer is not 624! Why do you think our answers might be off?

John Golden creates and implements a math game around decimal addition called Burger Time with some fifth graders:

Roll five dice to get ingredients for your burger. The numbers correspond to how many mm tall each part is.

Matthew Jones creates a Would You Rather? activity and one of his “defiant” students makes an effective justification of a counterintuitive choice:

He blurted out “I want to paint Choice C.” I told them at the beginning that there was no right/wrong answer, they just had to justify it. I was lucky enough that he thought of the reason why you’d want to paint the larger one. The only reason you’d want to paint the largest wall is because you are paid by the hour. It was really interesting watching him come to that conclusion. And to take pride and ownership in the way that he did.

Jennifer Wilson describes one of NCSM’s “Great Tasks” and shows how she gathers and sorts student work with a TI-NSpire.

Students are asked to create a fair method for cutting any triangular pizza into 3 equal-sized pieces of pizzas. I asked students to work alone for a few minutes before they started sharing what they were doing with others on their team. I walked around and watched.

I was in a small room recently with some futurists who were very excited about adaptive learning. The reasons for their excitement wouldn’t surprise you. “Prussian factory model of learning, learn at your own pace, et cetera.” I admit it all sounded very appealing and when I tried to articulate my frustration with their model, I didn’t get far at all. I sounded like just another rent-seeking teacher trying to preserve the outdated model that cuts his paycheck.

Futurists and math educators talk past each other. If I could jump into any futurist’s head and encode any particular understanding there to make dialog easier, it would be this:

Adaptive learning is like an iPod with infinite capacity and infinite capability to play any song ever recorded or sung, provided those songs were written by Neil Diamond.

If all you’ve ever heard in your life is Neil Diamond’s music, you might think we’ve invented something quite amazing there. Your iPod contains the entire universe of music. If you’ve heard any other music at all, you might still be impressed by this infinite iPod. Neil wrote a lot of music after all, some of it good. But you’ll know we’re missing out on quite a lot also.

So it is with the futurists, many of whom have never been in a class where math was anything but watching someone lecture about a procedure and then replicating that procedure twenty times on a piece of paper. That entire universe fits neatly within a computer-adaptive model of learning.

But for math educators who have experienced math as a social process where students conjecture and argue with each other about their conjectures, where one student’s messy handwritten work offers another student a revelation about her own work, a process which by definition can’t be individualized or self-paced, computer-adaptive mathematics starts to seem rather limited.

Lectures and procedural fluency are an important aspect of a student’s mathematics education but they are to the universe of math experiences as Neil Diamond is to all the other amazing artists who aren’t Neil Diamond.

If I could somehow convince the futurists to see math the same way, I imagine our conversations would become a lot more productive.

BTW. While I’m here, Justin Reich wrote an extremely thoughtful series of posts on adaptive learning last month that I can’t recommend enough:

Featured Comments:

Kent Haines:

Can I offer another analogy for these technologists? Adaptive learning is like a guitar teacher who teaches you how to play harder and harder pieces of music but never teaches you how to improvise. So you can play a piece of music that is placed in front of you, but you’ll never be able to pick up a guitar and just play with a couple of friends. I would contend that the improvisor is better prepared to understand and even make music. I’ll bet Neil Diamond can pick up a guitar and jam.

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