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.


Blind Trust

I attended Math for America’s Master Teachers on Teaching event last night. If you’re not familiar with it, it’s essentially a Ted-ED style event where math teachers give presentations on topics that they are passionate about. One of my biggest takeaways had to deal with technology and how I address how my students use it.

There were some awesome presentations and one that stood out to me was Patrick Honner’s talk about the shortcomings of technology as it relates to teaching and learning. A couple examples that Patrick pointed out were the TI-84’s inability to accurately represent continuity (asymptotes) and Desmos’s failure in representing holes in polynomial functions. Normally, technology allows beautiful, helpful representations of mathematics, but in all of these instances technology utterly fails to do this. Patrick encouraged the audience to embrace these types of pitfalls as teachable moments that could enable deeper understanding of the concepts.

This whole discussion reminded me of something that happened in my class a while back. Once during an exam, I witnessed a (precalculus) student enter “1 + 2” into their calculator as they solved a problem. In the moment, I almost laughed out loud (and afterwards I did). The next day when I handed back the exam, I brought my observation up to the class. The guilty student openly took ownership of the act and said:
“Mr. P, on an exam, you never know!

This example is slightly different than the ones Patrick highlighted, but it nonetheless brings attention to my students’ increasing dependency on technology. This connects with what was Patrick’s concluding point: our students blindly and inherently trust most all technologies they use – more so than they trust their own intuition. Combining this with technology’s pitfalls, and I see a recipe for disaster: students wholeheartedly trusting a flawed tool.

This means that I have a crucial responsibility to transform student reliance on technology into teachable moments that enable deeper understanding. It also means that if my students blindly trust technology and all it’s shortcomings and fail as a result, I have also failed.


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