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.

]]>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.

]]>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.

**Related Posts**

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!

]]>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.

**Related Posts**

- Regents Recaps
- Regents Recap — January 2015: It’s True Because It’s True
- Regents Recap — June 2016: What Do They Want to Hear?
- Regents Recap — June 2014: Common Core Algebra, “Explain your answer”

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First, number 15 from the June, 2017 Common Core Algebra exam.

This question puzzled me. The only unambiguous answer choice is (3), which can be quickly eliminated. The other answer choices all involve descriptors that are not clearly defined: “evenly spread”, “skewed”, and “outlier”.

The correct answer is (4). I agree that “79 is an outlier” is the best available answer, but it’s curious that the exam writers pointed out that an outlier would affect the standard deviation of a set of data. Of course, every piece of data affects the standard deviation of a data set, not just outliers.

From the Common Core Algebra 2 exam, here is an excerpt from number 35, a question about simulation, inference, and confidence intervals.

I can’t say I understand the vision for statistics in New York’s Algebra 2 course, but I know one thing we definitely don’t want to do is propagate dangerous misunderstandings like “A 95% confidence interval means we are 95% confident of our results”. We must expect better from our exams.

UPDATE: Amy Hogan (@alittlestats) has written a nice follow up post here.

**Related Posts**

What is the purpose of the artificial context? Why must the question be framed as though three people are comparing their answers? Why not just write a *math question*?This question not only addresses the same mathematical content, it makes the mathematics the explicit focus. This would seem to be a desirable quality in a mathematical assessment item.

Instead of wasting time concocting absurd scenarios for these problems, let’s focus on making sure the questions that end up on these exams are mathematically correct.

**Related Posts**

- Regents Recaps
- Not Even Psuedo-Context
- Regents Recap — January 2016: No It Wasn’t
- g = 4, and Other Lies the Test Told Me

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I’m excited to announce the launch of my column for Quanta Magazine! In *Quantized Academy* I’ll be writing about the fundamental mathematical ideas that underlie Quanta’s stories on cutting edge science and research. Quanta consistently produces exciting, high-quality science journalism, and it’s a tremendous honor to be a part of it.

My debut column, *Symmetry, Algebra and the Monster*, uses the symmetries of the square to explore the basic group theory that connects algebra and geometry.

You could forgive mathematicians for being drawn to the monster group, an algebraic object so enormous and mysterious that it took them nearly a decade to prove it exists. Now, 30 years later, string theorists — physicists studying how all fundamental forces and particles might be explained by tiny strings vibrating in hidden dimensions — are looking to connect the monster to their physical questions. What is it about this collection of more than 10^53 elements that excites both mathematicians and physicists?

The full article is freely available here.

]]>Here’s an animation I made to celebrate.

**Related Posts**

- 9/25/16 — Happy Pythagorean Square Day!
- 4/9/16 — Happy Square Day!
- 12/9/15 — Happy Right Triangle Day!

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The New York State Education Department has now admitted that at least three mathematically erroneous questions appeared on the June, 2017 Geometry exam. It’s bad enough for a single erroneous question to make it onto a high-stakes exam taken by 100,000 students. The presence of three mathematical errors on a single test points to a serious problem in oversight.

Two of these errors were acknowledged by the NYSED a few days after the exam was given. The third took a little longer.

Ben Catalfo, a high school student in Long Island, noticed the error. He brought it to the attention of a math professor at SUNY Stonybrook, who verified the error and contacted the state. (You can see my explanation of the error here.) Apparently the NYSED admitted they had noticed this third error, but they refused to do anything about it.

It wasn’t until Catalfo’s Change.org campaign received national attention that the NYSED felt compelled to publicly respond. On July 20, ABC News ran a story about Catalfo and his petition. In the article, a spokesperson for the NYSED tried to explain why, even though Catalfo’s point was indisputably valid, they would not be re-scoring the exam nor issuing any correction:

“[Mr. Catalfo]used mathematical concepts that are typically taught in more advanced high school or college courses. As you can see in the problem below, students weren’t asked to prove the theorem; rather they were asked which of the choices below did not provide enough information to solve the theorem based on the concepts included in geometry, specifically cluster G.SRT.B, which they learn over the course of the year in that class.”

There is a lot to dislike here. First, Catalfo used the Law of Sines in his solution: far from being “advanced”, the Law of Sines is actually an optional topic in NY’s high school geometry course. Presumably, someone representing the NYSED would know that.

Second, the spokesperson suggests that the correct answer to this test question depends primarily on what was supposed to be taught in class, rather than on what is mathematically correct. In short, if students weren’t supposed to learn that something is true, then it’s ok for the test to pretend that it’s false. This is absurd.

Finally, notice how the NYSED’s spokesperson subtly tries to lay the blame for this error on teachers:

“For all of the questions on this exam, the department administered a process that included NYS geometry teachers writing and reviewing the questions.”

Don’t blame us, suggests the NYSED: it was the teachers who wrote and reviewed the questions!

The extent to which teachers are involved in this process is unclear to me. But the ultimate responsibility for producing valid, coherent, and correct assessments lies solely with the NYSED. When drafting any substantial collaborative document, errors are to be expected. Those who supervise this process and administer these exams must anticipate and address such errors. When they don’t, they are the ones who should be held accountable.

Shortly after making national news, the NYSED finally gave in. In a memo distributed on July 25, over a month after the exam had been administered, school officials were instructed to re-score the exam, awarding full credit to all students regardless of their answer.

And yet the NYSED still refused to accept responsibility for the error. The official memo read

“As a result of a discrepancy in the wording of Question 24, this question does not have one clear and correct answer. “

More familiar nonsense. There is no “discrepancy in wording” here, nor here, nor here, nor here. This question was simply erroneous. It was an error that should have been caught in a review process, and it was an error that should have been addressed and corrected when it was first brought to the attention of those in charge.

From start to finish, we see problems plaguing this process. Mathematically erroneous questions regularly make it onto these high stakes exams, indicating a lack of supervision and failure in management of the test creation process. When errors occur, the state is often reluctant to address the situation. And when forced to acknowledge errors, the state blames imaginary discrepancies in wording, typos, and teachers, instead of accepting responsibility for the tests they’ve mandated and created.

There are good things about New York’s process. Teachers are involved. The tests and all related materials are made entirely public after administration. These things are important. But the state must devote the leadership, resources, and support necessary for creating and administering valid exams, and they must accept responsibility, and accountability, for the final product. It’s what New York’s students, teachers, and schools deserve.

**Related Posts**

- Regents Recaps
- The Worst Regents Question of All Time
- More Mathematical Misunderstanding
- Trouble with Dilations (And Logic)
- Another Embarrassingly Bad Math Exam Question

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