Get Posts by E-mail

Archive for the 'classroomaction' Category

2015 Remainders

Let’s close out 2015. In this remainders edition:

  • Eight new blog subscriptions from November & December.
  • Five essential 2015 posts from this blog.
  • Three bloggers I envy.
  • Seventeen Great Classroom Action posts I never got around to posting.


  • We successfully goaded Brett Gilland into tweeting and blogging. His writing features art, wit, and insight for days. Best follow of my fall quarter.
  • Jason D’Arcangelo is an elementary math coach, making him rare company online.
  • Kendra Lomax does interesting work in elementary math education also, most recently with the University of Washington’s Teacher Education by Design project.
  • Damian Watson just came off a two-year blogging hiatus with a post featuring Malcolm Swan, Andrew Stadel, and cognitive conflict, which pushes all three of my buttons.
  • Meryl Polak likewise came off a maternity leave to post about her experience designing and implementing a 3 Act Math task.
  • Geoff Wake was one of my colleagues at the Shell Centre when I set up a tent in their offices several years ago. Great guy. Interesting thinker. I’m excited to see him maintaining a blog.
  • Jenn Vadnais does consistently interesting work with the Desmos Activity Builder. I’m tuned in, hoping to learn how she works.
  • Glen Lewis blogs thoughtfully about technology, learning, and engagement in math education.

These blogs are each low volume, producing maybe one post per month. There is zero risk of getting overwhelmed here. Just toss them in Feedly or some other RSS reader and enjoy their insight whenever they find the time to share it.

Honorable Mentions

I don’t have a lot of envy in me for other Internet math ed types – their followers, retweets, subscribers, etc. Just keep working. What does turn me green, what I do covet, though, is another blogger’s ability to stir up conversation, to mobilize and collect the intellect of his or her readers. In 2015, that was Dylan Kane, the blogger whose posts invariably had me clicking through to the comments to see what he managed to provoke from his readers, then scratching my head trying to figure out how he did it.

If your heart belongs to elementary math education, the best moderators I have found there are Tracy Zager and Joe Schwartz.

My Year in Review

If you’ve come to this blog only recently, here are five posts that received a lot of traffic and commentary this year:

Looking for favorites from the wider online math education community? Check out the #MTBOS2015 hashtag. If I had to award my own MVP, it’d be Elizabeth Statmore’s “How People Learn” and how people learn where she turns essential research into manageable practice.

Great Classroom Action

And now, shamefully presented without commentary, seventeen posts I read in 2015 that had me check myself and think, “That classroom action is great!” I haven’t shared these yet and it’s time to clean the cabinet.

Great Classroom Action


Scott Keltner sent a drone up in the sky while his students plotted themselves down below. I emailed Scott and asked for his lesson plan and he sent back something involving playing cards and, frankly, none of it made any sense to me and I seriously don’t understand how the downside of cost, time, and effort could possibly outweigh the upside of drones (!) but I’m so curious. Bug Scott via Twitter to write up what he did here.

Julie Reulbach used zombies to create a need for logarithms. Zombies are, obviously, catnip for some students, but that isn’t what caught my eye here. Julie understood that logarithms are a shortcut for inverting an exponential equation. And if you’d like to create a need for a shortcut it’s helpful for students to experience the longcut, however briefly. Watch her work.

Ollie Lovell used one of my unposted tasks with a group of students in Myanmar who spoke very limited English and whose classroom had no electricity. Imagine how your favorite lessons would have to change under those constraints and then read how Ollie changed his. I learned a lot.

Sarah Hagan shares a game from S T called Greed, which helped turn her students’ perception of box-and-whisker plots from useless to useful. Crucially, the game exploits the need for box-and-whisker plots, which is comparison between multiple sets. Creating a box-and-whisker plot for a single set of data will feel pointless, same as teaching someone to use a carrot peeler by using it to paint a house. That’s not what it’s for!


Classkick allows you to give your students written feedback on work you assign on iPads. Crucially, that student work can be handwritten, which is (potentially) more valuable for feedback than multiple choice work. I thought it looked promising and I wrote about some of its promise last September.

Ruth Eichholtz didn’t find it as useful in class (where it’s hard to focus on a dashboard) as she did out of class, when she took a personal day:

I had my iPad at home and had iPads brought to the students at the beginning of the lesson. They were monitored at the start by a substitute teacher, who made sure they were present and that they received my email instructions. And then they joined my lesson on Classkick and worked, for 75 minutes, with me. As the students worked through each review problem, I could see their progress. I could make comments on their solution methods, correct their mistakes, and praise their successes. A few times, I tried to tell them they could use pencil & paper and just resort to Classkick when they needed help, but every single one chose to work on the iPad for the entire lesson!

I’m pessimistic about any vision of math education that has a robot grading the work of millions of students. These robots just aren’t good enough yet.

I’m intrigued, however, by this vision of math education that has one expert human analyzing and responding to the handwritten mathematical thinking of many more students than could fit in the same room at the same time. Let’s push ahead a little farther on that path.

Featured Comment


The first thing I thought about after seeing the tutorial was that I could start a mini Saturday school lesson for those students that need or want the help. Also, some of my students have a lot of after school activities, I have meetings and trainings, so I could see setting a time where we could review some work.


This goes beyond “great classroom action.” These are great moments that go great lengths towards defining the culture of a classroom.

Nathaniel Highstein offers a full week of activities:

The start of the school year is one of the most important moments for my classes. Setting the right tone and attitude right from the beginning can mean buy-in from students right away – and conversely, a bad start can be really tough to recover from. I had a pretty good start this year in my Algebra 1 and Algebra 2 classes. I wanted to share some things that worked for me in case someone else might benefit, and to document the week, as I may repeat much of this work next year.

You can find lots of posters that describe the practices of a mathematician, the elements of a growth mindset, etc. Sam Shinde went a different direction, turning student work into a poster, and annotating it with mathematical practices.

Nora Oswald uses a viral YouTube video to illustrate the mathematical practice of perseverance:

After we watch this, I like to make the connection to the classroom.
“Do you ever feel like you’re driving around in circles?”
“Do YOU ever feel like you look like a fool and others are laughing at you?”
“Did this woman give up even though she may have looked foolish and stupid?”
“At what point do you ask for help?”

As best as I can tell, Jonathan Claydon has constructed a positive and productive classroom culture entirely out of gags like this.

I spent a year working on Dandy Candies with around 1,000 educators.

In my workshops, once I stop learning through a particular problem, either learning about mathematics or mathematics education, I move on to a new problem. In my year with Dandy Candies, there was one question that none of us solved, even in a crowd that included mathematics professors and Presidential teaching awardees. So now I’ll put that question to you.

Here is the setup.

At the start of the task, I ask teachers to eyeball the following four packages. I ask them to decide which package uses the least packaging.


With the problem in this state, without dimensions or other information, lots of questions are available to us that numbers and dimensions will eventually foreclose. Teachers estimate and predict. They wonder how many unit cubes are contained in the packages. They wonder about descriptors like “least” and “packaging.”

After those questions have their moment, I tell the teachers there are 24 unit cubes inside each package. Eventually, teachers calculate that package B has the least surface area, with dimensions 6 x 2 x 2.

We then extend the problem. Is there an even better way to package 24 unit cubes in a rectangular solid than the four I have selected? It turns out there is: 4 x 3 x 2. When pressed to justify why this package is best and none better will be found, many math teachers claim that this package is the “closest to a cube” we can form given integer factors of 24.


The Problem We Never Solved

We then generalize the problem further to any number of candies. I tell them that as the CEO of Dandy Candies (DANdy Candies … get it?!) I want to take any number of candies – 15, 19, 100, 120, 1,000,000, whatever – and use an easy, efficient algorithm to determine the package that uses the least materials.

Two solutions we reject fairly quickly:

  1. Take your number.
  2. Write down all the sets of dimensions that multiply to that number.
  3. Calculate the packaging for that set of dimensions.
  4. Write down the set that uses the least packaging.


  1. Take your number.
  2. Have a computer do the previous work.

I need a rule of thumb. A series of steps that are intuitive and quick and that reveal the best package. And we never found one.

Here was an early suggestion.

  1. Take your number.
  2. Write down all of its prime factors from least to greatest.
  3. If there are three or fewer prime factors, your dimensions are pretty easy to figure out.
  4. If there are four or more factors, replace the two smallest factors with their product.
  5. Repeat step four until you have just three factors.

This returned 4 x 3 x 2 for 24 unit candies, which is correct. It returned 4 x 5 x 5 for 100 unit candies, which is also correct. For 1,000 unit candies, though, 10 x 10 x 10 is clearly the most cube-like, but this algorithm returned 5 x 8 x 25.

One might think this was pretty dispiriting for workshop attendees. In point of fact it connected all of these attendees to each other across time and location and it connected them to the mathematical practice of “constructing viable arguments” (as the CCSS calls it) and “hypothesis wrecking” (as David Cox calls it).

These teachers would test their algorithms using known information (like 24, 100, and 1,000 above) and once they felt confident, they’d submit their algorithm to the group for critique. The group would critique the algorithm, as would I, and invariably, one algorithm would resist all of our criticism.

That group would name their algorithm (eg. “The Snowball Method” above, soon replaced by “The Rainbow Method”) and I’d take down the email address of one of the group’s members. Then I’d ask the attendees in every other workshop to critique that algorithm.

Once someone successfully critiqued the algorithm – and every single algorithm has been successfully critiqued – we emailed the author and alerted her. Subject line: RIP Your Algorithm.


So now I invite the readers of this blog to do what I and all the teachers I met last year couldn’t do. Write an algorithm and show us how it would work on 24 or another number. Then check out other people’s algorithms and try to wreck them.

Featured Comment

Big ups to Addison for proposing an algorithm and then, several comments later, wrecking it.

2015 Sep 25

We’re all witnessing incredible invention in this thread. To help you test the algorithm you’re about to propose, let me summarize the different counterexamples to different rules found so far.

20 should return 5 x 2 x 2
26 should return 13 x 2 x 1
28 should return 2 x 2 x 7
68 should return 2 x 2 x 17
222 should return 37 x 3 x 2
544 should return 4 x 8 x 17
720 should return 8 x 9 x 10
747 should return 3 x 3 x 83
16,807 should give 49 x 49 x 7
54,432 should return 36 x 36 x 42
74,634 should give 6 x 7 x 1777

Next »