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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.

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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.

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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

Great Classroom Action


Coral Connor’s students created 3D chalk charts to demonstrate their understanding of trig functions:

As a showcase entry we spent several lessons developing the Maths of perspective drawings of representations of comparisons between Australia and the mission countries- income, death rates, life expectancy etc, and finished by creating chalk drawings around the school for all to see.

Malke Rosenfeld assigned the Hundred-Face Challenge – make a face using Cuisenaire Rods that up to 100 – and you should really click through to her gallery of student work:

Some kids just made awesome faces. Me: “Hmmm…that looks like it’s more than 100. What are you going to do?” Kid: “I guess we’ll take off the hair.”

One of my favorite aspects of Bob Lochel’s statistics blogging is how cannily he turns his students into interesting data sets for their own analysis:

Both classes gave me strange looks. But with instructions to answer as best they could, the students played along and provided data. Did you note the subtle differences between the two question sets?

Jonathan Claydon shows us how to cobble together a document camera using nothing more than a top of the line Mac and iPad.

Great Sam Shah Action


Sam Shah’s blog has been a veritable teaching clinic the last two weeks, more than filling his own installment of Great Classroom Action.

With Attacks and Counterattacks, Sam asked his students to define common shapes as best as they could – triangle, polygon, and circle, for instance. They traded definitions with each other and tried to poke holes in those definitions.

When the counter-attacks were presented, it was interesting how the discussions unfolded. The original group often wanted to defend their definition, and state why the counter-attack was incorrect.

Trade the definitions back, strengthen them, and repeat.

Sam created some very useful scaffolds for the very CCSS-y question, “If you have a shape and its image under a rotation, how can you quickly and easily find its center of rotation?”

This is an awesome exercise (inmyhumbleopinion) because it has kids use patty paper, it has them kinesthetically see the rotation, and it gives them immediate feedback on whether the point they thought was the center of rotation truly is the center of rotation. Simple, sweet, forces some thought.

Sam then pulls a move with a Post-It note that is a stunner, simultaneously useful for clarifying the concept of a variable and for finding the sum of recursive fractions:

Ready? READY? Flip. THAT FLIP IS THE COOLEST THING EVER FOR A MATH TEACHER. That flip was the single thing that made me want to blog about this.

Finally, Sam pulls a masterful move in the setup to his students’ realization that all the perpendicular bisectors of a triangle’s side meet in the same point. He has them first find those lines for pentagons (nothing special revealed) and quadrilaterals (nothing special revealed) before asking them to find them for triangles (something very special revealed).

There were gasps, and one student said, and I quite, “MIND BLOWN.”

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