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The Math Behind Gerrymandering and Wasted Votes — Quanta Magazine

The U.S. Supreme Court is currently considering a case about partisan gerrymandering in Wisconsin and Texas. One of the keys to the case is the “efficiency gap”, an attempt quantify the partisan bias in a given electoral map. For my latest article in Quanta Magazine, I explain and explore the efficiency gap using simple examples, and talk about some of the implications of this particular measurement.

Imagine fighting a war on 10 battlefields. You and your opponent each have 200 soldiers, and your aim is to win as many battles as possible. How would you deploy your troops? If you spread them out evenly, sending 20 to each battlefield, your opponent could concentrate their own troops and easily win a majority of the fights. You could try to overwhelm several locations yourself, but there’s no guarantee you’ll win, and you’ll leave the remaining battlefields poorly defended. Devising a winning strategy isn’t easy, but as long as neither side knows the other’s plan in advance, it’s a fair fight.

Now imagine your opponent has the power to deploy your troops as well as their own. Even if you get more troops, you can’t win.

The full article is freely available here.

Global Math Week Symposium

James Tanton is on a mission to bring joyous mathematics to the world.  His Global Math Project is about to launch Global Math Week:  during the week of October 9th, over 600,000 students from around the world will enjoy a shared mathematical experience based on Tanton’s Exploding Dots, a wonderful, surprising, and awe-inspiring take on place value.

James has been traveling the world for the past year spreading the good word about mathematics and his exploding dots.  If you haven’t yet signed up, I encourage you to do so.  The mathematics is wonderful, relevant, and inspired, and the Global Math Project has lots of resources at their homepage.

To kick off Global Math Week, the Global Math Project together with the Museum of Mathematics will be hosting a symposium at NYU’s Courant Institute.  Mathematical luminaries like Po-Shen Loh, Henry Segerman, and many others will be on hand to celebrate.  And I’m honored to be participating in a panel discussion on Uplifting Mathematics for All, where we will discuss how to make mathematics meaningful, fun, and coherent in and out of the classroom.

So get ready for Global Math Week!  Hopefully this is the first of many to come.

Regents Recap, August 2017: How Do You Explain that Two Things are Equal?

Sue believes these two cylinders from the August, 2017 New York Regents Geometry exam have equal volumes. Is Sue correct? Explain why.

Yes, Sue, you are correct: the two cylinders have equal volumes. I computed both volumes and clearly indicated that they are the same. Take a look!

Wait. Why did I only get half-credit? What’s the problem, Sue? You don’t think this is an “explanation”? The two volumes are equal. The explanation for why they are equal is that I computed both volumes and got the same number. I don’t know of any better explanation for two things being equal than that.

What’s that? You wanted me to say “Cavalieri’s Principle”? But if I compute the two volumes and show that they are equal, why would I need to say they are equal because of some other reason?  Oh, never mind, Sue. See you in Algebra 2.

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AMS — Math in the Media

The debut of my column in Quanta Magazine was recently featured by the American Mathematical Society’s Math in the Media!

In addition to a nice review of my first Quantized Academy column, “Symmetry, Algebra, and the Monster“, I was also interviewed by Math in the Media’s Rachel Crowell.  Here’s an excerpt:

AMS: What excites you most about Quanta’s addition of the Quantized Academy series?

PH: Quanta does a wonderful job showing how mathematics and science are vibrant, active endeavors.  The writers bring math and science alive, telling exciting stories of mathematicians, scientists and their work. Quantized Academy can help connect students, teachers, and other lifelong learners to those stories and the math behind them.

You can read the entire article here.  Thanks to the AMS, and to Rachel Crowell, for taking an interest and helping to spread the word!

Regents Recap, June 2017 — Assessing Irrationality

Despite its shortcomings, this kind of question keeps appearing on New York State math exams.  This is number 27 from the June, 2017 Common Core Algebra exam.

Here’s an example of a full credit response according to the official model response set provided by the state.

There is no explanation here.  The argument is simply It’s True Because It’s True:  the difference between a rational number and an irrational number is irrational because the difference between a rational number and an irrational number is irrational.  All the student has done is identified one number as rational and one number as irrational (without even identifying which is which) and recited the frequently-tested property.

As scored, this question is designed to test recall of a specific, incidental fact while intentionally avoiding the relevant mathematical content, namely, what it means for a number to be rational or irrational.  A second model response that actually demonstrates some mathematical knowledge about irrational numbers earns only partial credit.

Unlike the student in the first response, or the test makers for that matter, the student here recognizes that the irrationality of the square root of 2 should be established.  The explanation isn’t completely correct, but it demonstrates much more understanding than the first response.  Unfortunately, as long as questions like this keep appearing on these exams, students and teachers will continue to be rewarded for mindlessly regurgitating what the test makers want to hear.

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