What Will They Be Doing?

When planning a lesson, start with the question “What will the students be doing?”

I received this piece of advice as a pre-service teacher and it has stuck with me my entire career.

Before you get too excited, my answer to this question is usually pretty straightforward; most often, it’s “Working on a problem I’ve shared” or “Thinking about a question I’ve asked“. But making this activity explicit in the planning process reminds me to focus on those things that matter most in my lesson: the specific questions I want to ask and the specific problems I want students to engage with.

Asking myself this simple question also helps keep the focus of my planning where it belongs. Instead of starting from “How do I understand this?”, I start from “How will my students come to understand this?” This is a small shift that makes a big difference.

 

NCTM Annual — 2018

I’m excited to be heading to Washington, DC in April for the 2018 NCTM Annual Meeting!

NCTM’s annual meeting brings together thousands of educators from across the country to discuss mathematics, pedagogy, technology, and more. I presented at the 2017 Annual Meeting in San Antonio and had a great time, so I’m looking forward to this year in DC.

I’ll be presenting Statistics and Simulation in Scratch, a 60-minute session about using simple computer programming tools to make the study of probability and statistics more experimental and exploratory. We’ll look at ways teachers and students can use Scratch, the free, web-based programming environment designed by the MIT Media Lab, to model simple probability experiments, collect and analyze data, and create mathematically compelling projects. The technology tools we’ll be using are free and intuitive, and they open up a new pathway to probability and statistics for students and teachers. In addition, it creates opportunities to learn and apply fundamental computer programming skills in a meaningful context.

My talk is scheduled for Thursday, 4/26/18, at 3:00 pm, so if you’re planning on attending the NCTM Annual, please keep my session in mind!

Conferences like this are great opportunities for professional growth, but the logistics are often complicated for classroom teachers.  I’m fortunate to have received support from Math for America, which makes attending NCTM’s Annual Meeting in Washington DC possible. And I’m proud to be one of several MfA teachers presenting at NCTM! You can find a complete list of MfA presenters here.

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MfA Workshop — Exploring Modern Discoveries in Mathematics and Science

This week I will be co-facilitating a workshop for teachers, “Exploring Modern Discoveries in Mathematics and Science”, with Thomas Lin, editor-in-chief of Quanta Magazine. We will be running the workshop for a group of Math for America math and science teachers at the MfA offices.

In our workshop we’ll look at ways to connect students and teachers with modern science research and discoveries. We’ll focus on resources from Quanta Magazine, including recent reporting on advances in mathematics, biology, and computer science, as well as some of my Quantized Academy columns.

I’m excited to be working with Tom, who in addition to being the founding editor of Quanta, is also a former teacher. Tom’s desire to make the amazing work being done by Quanta’s journalists and writers more accessible to teachers and students led to the development of my Quantized Academy column last year.

Be sure to check out Quanta Magazine, and you can find my Quantized Academy articles here.

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sin(x) + cos(x)

Here is a fun little exploration involving a simple sum of trigonometric functions.

Consider f(x) = sin(x) + cos(x), graphed below.

Surprisingly, it appears as though sin(x) + cos(x) is itself a sine function. And while its period is the same as sin(x), its amplitude has changed and it’s been phase-shifted. Figuring out the exact amplitude and phase shift is fun, and it’s also part of a deeper phenomenon to explore.

Consider the function g(x= a sin(x) + b cos(x). Playing around with the values of a and is a great way to explore the situation.

On the way to a complete solution, a nice challenge is to find (and characterize) the values of a and b that make the amplitude of g(x) equal to one. It’s also fun to look for values of a and b that yield integer amplitudes: for example, 5sin(x) + 12cos(x) has amplitude 13, and 4sin(x) + 3cos(x) has amplitude 5.

Ultimately, this exploration leads to a really lovely application of angle sum formulas. Recall that

sin(A + B) = sin(A) cos (B) + sin(B) cos(A)

If we let Ax, we get

sin(x + B) = sin(x) cos(B) + sin(B) cos(x)

With a little rewriting, we have

sin(x + B) = cos(B) sin(x) + sin(B) cos(x)

which looks similar to our original function f(x) = sin(x) + cos(x), except for what’s in front of sin(x) and cos(x). We handle that with a clever choice of B.

Let B = \frac{\pi}{4}. Now we have

sin(x + \frac{\pi}{4}) = cos(\frac{\pi}{4})sin(x) + sin(\frac{\pi}{4})cos(x)

sin(x + \frac{\pi}{4}) = \frac{\sqrt{2}}{2} sin(x) + \frac{\sqrt{2}}{2}cos(x)

And a little algebra gets us

sin(x) + cos(x) = \sqrt{2}sin(x + \frac{\pi}{4})

And so sin(x) + cos(x) really is a sine function! Not only does this transformation explain the amplitude and phase shift of sin(x) + cos(x), it generalizes beautifully.

For example, consider 5sin(x) + 12cos(x). We can rewrite this in the following way.

5sin(x) + 12cos(x) = 13 ( \frac{5}{13} sin(x) + \frac{12}{13} cos(x))

5sin(x) + 12cos(x)= 13 ( cos(\beta) sin(x) + sin(\beta) cos(x))

5sinx + 12cosx = 13 sin (x + \beta)

where \beta = arcsin(\frac{12}{13}) = arccos(\frac{5}{13}).

There’s quite a lot of trigonometric fun packed into this little sum. And there’s still more to do, like exploring different phase shifts and trying the cosine angle sum formula instead. Enjoy!

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My PAEMST Story

My heart sank a little as I watched the video.

It had been three months since I submitted my application for the Presidential Award for Excellence in Mathematics and Science Teaching (PAEMST). As I often do in the summer, I was reviewing materials from the school year, cleaning up and getting organized. I found my PAEMST application folder, noticed the video — the centerpiece of the application portfolio — and clicked play.

I remembered feeling pretty good about the lesson I chose to record. Re-watching it months later confirmed that it was a good lesson. But it wasn’t flashy. I didn’t perform a rap about even and odd functions. Students weren’t chasing each other around the classroom in a relay race. I didn’t dramatically slice a melon in half with a meat cleaver. My chance of earning the country’s highest honor for K-12 STEM teachers hinged on this video, but it was just me teaching a normal lesson. I resigned myself to the fact that my odds probably weren’t very good.

But as I continued to watch the video, my attitude slowly changed. No, it wasn’t flashy — my lessons never are — but it was a really good lesson. It was well-designed, well-executed, and well-received. Students were deeply engaged in complex mathematics. There was a clear arc that everyone could connect with. By the end of the video, my resignation had turned to pride: This is what happens every day in our classroom. This is who I am as a teacher. This was a normal day: exactly the right way to represent myself and my work.

I know what kind of teaching captures the public’s interest, and this wasn’t it. But I was proud of what was showcased in the video, even if it might not look like “great teaching” to an outside observer. Would PAEMST reviewers appreciate the well-chosen problems that bridged prior knowledge and new concepts? Would they notice the classroom culture in which students immediately began collaborating, seeking each other’s validation before mine? Would they see the subtle changes I made after assessing small group discussions? Would they appreciate how I strategically answered some questions and respectfully put others right back to the students? Would they notice how students listened to each other during whole-class discussion? How they comfortably responded to each other’s questions? How they made conjectures that would be resolved later in the lesson?

I guess they did.

I received the Presidential Award in 2013. Here I am, between then-US CTO Megan Smith and Dr. France Cordova, Director of the National Science Foundation. It was a tremendous honor to win the PAEMST and to travel to Washington D.C. to meet leaders from the National Science Foundation, the National Academy of Sciences, and the White House. And meeting and connecting with other awardees — teachers doing great work in all manners of classrooms, schools, and communities across the country — continues to impact the work I do.

And it was encouraging to know that those responsible for awarding the PAEMST understood what they were looking at when they watched my video: nothing flashy, just good teaching. The kind that happens in my classroom, and countless others around the country, every day. Years later, I still occasionally look at my PAEMST application materials: the essays, the artifacts, even the video. It’s a nice snapshot of where I was at in 2013, and it’s fun and productive to think about the ways I’ve changed, and stayed the same, as a teacher.

Creating that snapshot is one of the many reasons I encourage teachers to apply for the PAEMST. The application process is a worthwhile professional experience in and of itself. It’s the kind of work good teachers want to do anyway: planning instruction; thinking about curriculum; analyzing outcomes; reflecting on process. Applying for the Presidential Award is a great motivator to do that work.

Teachers out there who feel ready should consider applying. And if you know a great teacher, you can nominate them for the Presidential Award. The process alone is worth it, and the potential reward is career-changing.

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