Teaching

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.

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 b 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 A = x, 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!

Resources Teaching

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.

Teaching Testing

Regents Recap — January 2018: How Do You Explain That 2 + 3 = 5?

This has quickly become my new least-favorite kind of Regents exam question. (This is number 32 from the January, 2018 Algebra 2 Regents exam.)

What can you say here, really? They’re equal because they’re the same number. Here’s a solid mathematical explanation. Right?

Wrong.

According to those who write the scoring guidelines for these exams, this is a justification, not an explanation. Because students were asked to explain, not justify, this earns only half credit.

This is absurd. First of all, this is a perfectly reasonable explanation of why these two numbers are equal. This logical string of equalities explains it all. This clear mathematical argument demonstrates what it means to raise something to the power 3/4.

Second, whatever it is that differentiates an “explanation” from a “justification” in the minds of Regents exams writers, it’s never been made clear to test-takers or the teachers who prepare them. A working theory among some teachers is that “explain” just means “use words”. Setting aside how ridiculous this is, if this is the standard to meet, students and teachers need to be aware of it. It needs to be clearly communicated in testing and curricular materials. It isn’t.

Third, take a look at what the the test-makers consider a “complete and correct response”.

In this full-credit response, the student demonstrates a shaky mathematical understanding of the situation (why are they using logarithms?) and writes a statement (“81 with four roots gives you 3”) that, while on the right track, is in need of substantial mathematical refinement. Declaring this to be a superior response to the valid mathematical argument above is an embarrassment.

These tests are at their worst when they encourage and propagate poor mathematical behavior. We deserve more from our high-stakes exams.

Teaching Testing

Regents Recap — January 2018: Is it Better to Justify or Explain?

On question 32 of the January, 2018 Common Core Algebra 1 Regents exam, students were asked to explain why a quadratic whose graph is given might have a particular set of factors. Here are two sample student responses from the state-produced Model Response Set.

On the left, the student says “Yes”, sets each factor to 0 and solves, and produces the roots x = -2 and x = 3. On the right, the student says “Yes, because the x-intercepts are (-2,0) and (3,0).”

One of these responses received full credit, the other half credit. I posted this to Twitter and invited people to guess.

One of these responses got full credit, the other got half credit. Which is which?#math #mathchat pic.twitter.com/4fw1QK2GWP

— Patrick Honner (@MrHonner) February 11, 2018

According to the official scoring guide, the response on the right earned full credit: it is a “complete and correct response”. The response on the left earned half credit, because the student “gave a justification, not an explanation.”

It seemed as though the majority of respondents on Twitter favored the response on the left; a few even specifically said it offered a better “explanation” than the full-credit response. Many did choose the response on the right, especially those familiar with how New York’s Regents exams are scored.

To me, both answers are unsatisfying. The full-credit response offers an “explanation” but is devoid of justification: the student doesn’t make the connection between the x-intercepts and the roots. The half-credit response derives the roots algebraically, but fails to explicitly connect the roots to the intercepts. It’s hard for me to accept that one of these responses is substantially better than the other: both responses expect the reader to fill in an equally important gap.

It’s also hard for me to accept what counts as “explanation” here. Several teachers familiar with New York’s Regents exams commented that, in this context, “explain” simply means use words. And we’ve seen example after example of ridiculous “explanations” on these exams. It sends the wrong message to students and teachers about what constitutes mathematics, and since the message is transmitted via high-stakes exams, it can’t be ignored.

MfA Workshop — Exploring Modern Discoveries in Mathematics and Science

sin(x) + cos(x)

My PAEMST Story

Regents Recap — January 2018: How Do You Explain That 2 + 3 = 5?

Regents Recap — January 2018: Is it Better to Justify or Explain?

Follow Mr Honner