PD with Dan, modeling with mathematics

With the help of the NYCDOE, this past Wednesday was the third of three workshops that I’ve attended with Dan Meyer this school year. I’ve written about the first two, so why not cap off the trilogy? Read more here and here. In an effort to solidify my experience and have something to reflect upon at a later date, here’s a recap of the session.

The focus was on mathematical modeling. This is one of my weaknesses, so when I walked in and looked at the agenda, I was pretty happy. Teach me Dan! Things opened up with us sharing what we thought it meant for students to model with mathematics. The answers varied, but after feeling us out, Dan emphasized the difference between model the noun and model the verb. I had to catch myself here too as I hadn’t played close enough attention to this subtly. He used Desmos to collect our responses and our comfort level with mathematical modeling.

We then dove into the Penny Pyramid 3-Act. Dan worked his magic, as usual. During the debrief, he pulled from the CCSS and presented five actions that are necessary for students to be part of modeling experiences:

  • Identify variables
  • Formulate models
  • Perform operations
  • Interpret results
  • Validate conclusions

We talked about how the Penny Pyramid, while probably not ideal, works to get students to do all of these things. Our questions helped us to identify variables. Interestingly, Dan highlighted the vocabulary that we chose to represent the variables. This is important because, indirectly, he demanded precision and consistency from all of us when it came to describing key parts of the experience. Our tabular, algebraic, and physical representations of the pennies were the models, which we used to calculate the number of pennies in the pyramid. We spent time as a group discussing the meaning of our calculations and parts of our models – even those that weren’t correct. For example, one group landed on 22,140 pennies, which didn’t take into account that each stack was 13 pennies. Also: what does the 612 mean? The weakest part of our models was the validation. It’s not like we could actually count all the pennies. That said, we would compare outcomes of different models. There was also a newspaper article on the pyramid which helped with the actual count of pennies. Validation runs on a spectrum.

The afternoon was spent creating and improving some modeling tasks. We started with 25 billion apps and then went into a few graphing stories. We spent time thinking about how to structure modeling activities using these tasks. A great conversation occurred while exploring Distance from Camera. Someone brought up whether the graph should be piecewise linear or rounded periodic (like the solution). It was a great discussion, worthy of its own post. Dan remained vulnerable and eloquently showed us this ferris wheel Geogebra applet from the awesome John Golden. (Check out his full collection of Geogebra resources.)

We brainstormed how technology can help make our modeling dreams come true. The list was plentiful. Staying away from “textbook traps” topped the list. These include giving up numbers, tools, and other key information far too early in the modeling process. It’s also ironic that, because of technology, Dan was behind a desk for extended periods of time while we worked (he was typing our insights and using Desmos). Without technology, this teacher move is frowned upon.

The culminating task had us dig into several different versions of Barbie Bungee. Having never actually done this modeling activity, I learned not only about how to do, but also what not to do. Truth be told, if I would have looked at the activities before the workshop that day, I wouldn’t have suggested any changes. I would have been excited to implement them as is. But after purposefully rethinking modeling with mathematics with Dan for five hours, I felt very different.


  • The Penny Pyramid task would serve as a great introduction to summation notation.
  • Dan: “When there’s a great classroom experience, I ask myself: how could I have ruined this?
  • Not every aspect of modeling needs to every lesson. The goal is to feed students a healthy diet of modeling verbs.
  • Focus on broad questions. These will lead to more specific, granular questions. Avoid the reverse.
  • Seeing every teacher move I make as an investment into the lesson. Which moves are worth their investment? Which aren’t? This reflects the gravity that every decision we make. I need to be more deliberate.
  • Depending on what info we give kids, the level of modeling that they do could be very different. Don’t do the modeling for them! Give them procedural stuff.
  • It’s not if to give a handout, but when. This outcome was directly tied to Dan’s first session back in December, which helped things come full circle.
  • Once again, I was impressed with how Dan managed the audience. There were so many slick teaching moves (see notes).
  • I loved catching up with Sahar during lunch. We talked about her experiences visiting students homes with her school. She also gave me a tip about taking photos while students are working during the first couple of days of school to showcase how mathematical discussions should happen amongst students.
  • This wonderful periodic modeling Desmos activity.
  • If I ever do Barbie Bungee, I need to use Dan’s intro video.
  • Be mindful when moving between the real and math worlds. Don’t get lost on your travels.

Dan’s Google Doc for the session.


PD with Dan, creating intellectual need

This past week was workshop #2 with Dan Meyer, who was invited by the NYCDOE to conduct a three-part PD series. It’s not every day that you get to attend a workshop with him, so I’ve decided to capture each of my experiences. More on workshop #1.

Too make a long story short, it was another outstanding experience. Dan Meyer never ceases to be thought-provoking. He presents the sort of stuff that stays with you long after you get back to your classroom. Here are the details.

The focus of the session was creating intellectual need in our classrooms. He’s advocating for the learning of mathematics that doesn’t need to real world, job world, or even related to student interest. His argument was based on the work of Guershon Harel (article here). Dan believes, and I do too, that we can meaningfully engage our students by creating headaches in our classroom for our students. We shouldn’t be in the business of imposing mathematics where it isn’t wanted or welcomed. I’ve read his headache-inspired posts from his blog, so it was really cool to get to experience first hand his thoughts on the matter.

He hooked us by running through a series of small activities that are examples of creating intellectual need.

  • Little expressions. This creates a need for combining like terms and efficient calculation techniques.
  • Controversy. This creates a need a need for how we communicate and represent mathematics a series of operations.
  • Memory Game. He flashed first a 9-digit number and then a 16-digit number and asked us to remember as many digits as possible. Created a need for scientific notation, an efficient means of mathematical communication.
  • The $20 Bet. He wrote down a number and gave a volunteer ten attempts as guessing what it was. If they could guess his number, he’d give them $20 (he only had $5 though). After each guess, he let the volunteer know if their guess was too high or too low. His number ended up being 87.21! This creates a need for different number families and their relevance.
  • Parallel Lines. Creates a need for precision when calculating and representing two parallel lines. The coordinate plane rescues us.

We then explored two of Guershon Harel’s five components of intellectual need: the need for computation and the need for communication. These two needs were directly tied to the five activities that Dan shared. More on the five intellectual needs here.

Dan mentioned that there are three questions that he asks himself when he attempts to design experiences that create intellectual need.

  1. If [x] is the aspirin, then what’s the headache?
  2. Why did mathematicians invent [x]? Can I put students in that place even for a moment?
  3. How can I help students view [x] as powerful, not punishment?

A common theme throughout the day was how we should get into the habit of turning up the dial slowly. You can always give more information to your class, but you can never take it away. SO TRUE. This connected well with session 1, specifically the use of the white rectangle to remove information and increase access. The introduction to a lesson (the Do Now) was emphasized as a critical phase of creating intellectual need – students must be able access the content however inefficient their means may be.

The afternoon began with an activity creating a need for proper labeling and name-giving in geometry. Dan had a bunch of random points on the screen and had two volunteers each choose one and attempt to describe which point they’d chosen to the other person. Another headache ensued. For the 2nd person, he labeled the points with A, B, C, … and the aspirin was given.

We then were broken up into groups and were given a scenario. They all showcased the opposite of what a needs-based classroom looks like. We were asked perform a makeover. We jigsawed it back together, read the summary of each prescribed remedy from Harel, and everyone in our original group shared. What stemmed from the conversation was awesome: developing a need for the algebraic form of a function. The Points Desmos activity followed, emphasizing the usefulness of inequalities when representing all points that satisfy a given set of conditions.

Lastly, we explored Polygraph as a way of creating a need for math-specific language related parabolas. This was great.

Other interesting bits:

  • The word student means “to take pain” in some language (can’t remember which)
  • Whenever students laugh during an activity, you know their pain has been relieved
  • Algebra is sophisticated version of trial and error
  • Math pedagogy aside, I’m always find it compelling how Dan manages his audience. He greets everyone at the door. His warm use of “friends” and “colleagues” whenever referencing the audience makes everyone feel a sense of togetherness despite being strangers. I also liked his use of the phrase “For those of you that have the answer, say it out loud.”
  • After discussing the need for the algebraic representation of a function, Dan referred to algebra is “a more sophisticated form of trial and error.”

Dan’s Google Doc of the session.


PD with Dan, diagnosing the paper disease

Last week I attended a workshop led by Dan Meyer, hosted by the NYCDOE. This was the first in a series of three that I’ll be fortunate enough to attend with him this school year.

The focus of the session was to diagnose what Dan referred to as the paper disease. It’s the idea that learning mathematics through paper (like a textbook, for example) restricts not only how students learn mathematics, but also how they’re thinking about mathematics.

He demonstrated ways to use technology to open up problems to a wider audience of students. Of course Desmos was a focal point, but his oh-so simple method of using white rectangles in Keynote me struck me even more.

Here’s how it works: take a problem, any traditional problem typically found on a state exam or textbook, and screenshot it into a presentation software (keynote, PowerPoint, whatever). Start removing information given by covering up some of the info in the problem with a white rectangle. Repeat this process until you have something that can spark curiosity and give access to a far wider range of students. You’re basically deleting part (or most) of the problem, which may include the question objective itself. Less information equals greater access; it allows for students to formulate questions and make inferences about the info in the problem before even attempting to answer it.

The other huge takeaway for me was his development of informal v. formal mathematics. This could be interpreted as meeting students where they are, but I feel that it’s much more than that. Getting kids to think informally about mathematics during a lesson – especially at the beginning – requires far different planning than simply leveraging prerequisite knowledge. It’s more about how students are engaging with mathematics rather than whatever content they already know. Informal math also feels a hell of a lot different than formal math. When students are immersed in informal mathematics, they don’t even realize they’re doing mathematics. The same can’t always be said for formal mathematics.

Closing the loop, Dan argued that learning mathematics through paper flattens informal mathematics onto formal mathematics…instead of using one as a bridge to the other. This act injects our students with the paper disease.

I left the workshop wondering about how I’ve made math a highly formalized routine for my students. I left wondering how I would begin using the white rectangle. I left wondering about the unit packets that I create for my students, that together form my own textbook and how they’re impacting my students learning of math. I left wondering about the power of estimation. I left wondering how less is actually more.

Dan’s Google Doc of the session.


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