These cacti caught my. I can see both a dodecagon and a star in the 12-fold symmetry of the cactus in front. And to my surprise, the cactus behind it has thirteen sections!

I wonder about the range, and deviation, of the number of sections of these cacti. And what are the biological principles that govern these mathematical characteristics?

]]>For example, when proving an algebraic identity, multiplying both sides of an equation by 2 is permissible, because *x *= *y* and *2x *= *2y* are true under exactly the same conditions on *x* and *y*. Squaring both sides of an equation however, is not, since

can be true under conditions that make *x *= *y* false, say, when *x *= *2 *and *y* = *-2*.

The post in question, “Algebra is Hard”, was a response to a June 2016 Regents scoring guide that deducted a point from a student who, in proving an algebraic identity, multiplied both sides of the equation by a non-zero quantity. The student was penalized for “*not manipulating expressions independently in an algebraic proof*“, a vague and meaningless criticism.

“Algebra is Hard” received quite a bit of attention, and while many agreed with me, I was genuinely surprised at how many readers disagreed. Which was terrific! Of course my argument makes perfect sense to me, but it was great to have so many constructive conversations with teachers and mathematicians who saw things differently.

But my argument recently received support from the most unlikely of sources: another Regents exam.

Take a look at this exemplar full-credit student response to an algebraic identity on the August 2017 Algebra 2 exam.

Notice that the student works on both sides of the equation and subtracts the same quantity from both sides. Even though the student did not *manipulate expressions independently in an algebraic proof*, full credit was awarded.

The note here about domain restrictions is an amusing touch, given that it was the explicit domain restriction in the problem from 2016 that ensured the student wasn’t doing something impermissible (namely, multiplying both sides of an equation by 0).

So in 2016 this work gets half credit, and in 2017 this work gets full credit.

While it’s nice to see mathematically valid work finally receiving full credit on this type of problem, it’s no consolation to the many students who lost points for doing the same thing the year before. What’s especially frustrating is that, as usual, those responsible for creating these exams will admit no error nor accept any responsibility for it.

Be sure to read “Algebra is Hard” (and some of the 40+ comments!) for more of the backstory on this problem.

**Related Posts**

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”

]]>

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

]]>

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.

]]>