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.

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Here’s another lovely mathematical encounter from the New York Botanical Garden, and a nice companion piece to *Spiky Symmetry*. I believe this is known as a balloon cactus (*Paradoia Magnifica*). With 32 ribs, I can’t help but wonder how high the rotational symmetry group can go!

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We start by thinking about how rumors spread.

Let’s say you hear a juicy rumor that you just can’t keep to yourself. You hate rumormongers, so you compromise by telling only one person and then keeping your mouth shut. No big deal, right? After all, if the person you tell adopts the same policy and only tells one other person, the gossip won’t spread very far. If one new person hears the rumor each day, after 30 days it will have spread to only 31 people, including you.

So how bad could it be to tell two people? Shockingly bad, it turns out. If each day, each person who heard the rumor yesterday tells two new people, then after 30 days the rumor will have reached more than a quarter of the world’s population (2,147,483,647 people, or 231 − 1, to be exact). How can such a seemingly small change — telling two people instead of one — make such a big difference? The answer lies in rates of change.

A similar mathematical model can be used to understand the spread of disease. And by unpacking the mathematics behind the *basic reproduction number *of a disease, we can compute the critical cutoff for herd immunity.

Learn more by reading the full article, which comes with a classroom-ready worksheet and is freely available here.

]]>Nick diagrammed it out for me.

“Cherry pie is delicious” –> –>

“Apples pies are, too” –> –>

Now, I don’t mind a good mnemonic now and then; I still sing the alphabet song, after all. But this struck me as extremely silly. These formulas get used all the time and they are deeply connected to many other important concepts. Relying on a memory trick creates a flimsy foundation for an important body of knowledge. I decided to show Nick just how flimsy.

The next day in class, I approached Nick. “You know, after thinking about it, I agree with you: apple pies are delicious.” He was pleased. But his smile quickly receded. He wrote something out in his notes. “Wait, that’s not right.”

“So apple pies are not delicious?” I asked.

“It’s ‘*Cherry pie is delicious*‘.” He showed me the formula.

“But apples pies are delicious, right?”

“Yeah, but that’s just not how it works.”

“This is kind of confusing”, I said. “Oh wait. Now I see. Apple pies are delicious too!” I wrote out , followed by . “Perfect!”

“What?”

“See here,” I said. I wrote out . “You’re method works perfectly!”

Nick started scribbling more in his notebook. Having maximized confusion, I walked away.

Over the next few days I continued my demonstration. “Cherry pies are delicious, too!” I’d say. Or, “Apple pies are really, really delicious!” I might have even said something like “Some apple pies are to die for.”

My demonstration was successful. Maybe too successful. Nick got the area of circle wrong on the next test.

When I handed it back, he acknowledged my point with a combination of irritation and admiration. Nick never got the area of a circle wrong again. And we never had to talk about his Dear Aunt Sally again, either.

[*No mathematical understanding was harmed in this story*.]

I’m looking forward to meeting students and teachers, visiting some classes, learning more about Scarsdale HS, and talking about tilings!

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Celebrate Permutation Day by mixing things up! Try doing things in a different order today. Just remember, for some operations, order definitely matters!

]]>One of the biggest events for me this year was the launch of my column for Quanta Magazine. In *Quantized Academy* I write about the fundamental mathematical ideas that underlie Quanta’s stories on cutting edge science and research. This past year I’ve published columns on symmetry and group theory, gerrymandering, and pentagonal tilings, and some of my pieces have also appeared at *Wired*. It’s been a great experience (and challenge!) so far, and I’m looking forward to seeing where it goes in 2018.

I also continue to contribute to the NYT Learning Network (like this piece on gerrymandering) and have kept up the tradition of writing about the New York State Math Regents exams in 2017, which included one of the worst Geometry tests I’ve ever seen.

I also read a lot of books this past year, in an attempt to find healthier ways to spend my time, and I posted a list of some of the most interesting things I read in 2017. And thanks to a terrific mini-course on data representation with Mona Chalabi, I was inspired to create this Year in Math graphic.

I’ve continued to work to integrate mathematics and computer science in my classroom. This school year I’ve begun piloting a Mathematical Computing course, which is in part based on the work I’ve been doing in Scratch the past few years. I’ve presented about this work at Math for America, the NCTM Annual Meeting, and I’ve been featured by the Scratch Ed community. I plan on continuing to promote new work this year at similar venues.

I had another busy year speaking about mathematics, teaching, and technology. In addition to presenting at conferences like the NCTM Annual Meeting, I delivered the opening keynote at the inaugural MfA Summer Think conference, spoke at a STEM Grand Challenges event hosted by 100kin10, and participated in a panel discussion at the Global Math Week Symposium. I also designed and ran an interactive exhibit at the 2017 World Science Festival. Perhaps my biggest speaking honor this past year was keynoting Math for America’s annual Fall Function, where Giselle George-Gilkes and I spoke to 1,600 teachers and guests about the impact MfA has had on our careers. I already have a lot planned for 2018, but those with speaking inquiries can contact me here.

As always, I’m thankful to be able to reflect on a fulfilling professional year, and I look forward to another good one in 2018.

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This workshop is a product of my ongoing efforts to integrate mathematics and computer science in my classrooms. The study of probability creates natural opportunities to bring in tools from computer science, which create alternate pathways to understanding concepts in probability through generating, managing, and analyzing data.

I will also be presenting on this topic at the NCTM Annual Meeting in Washington, DC in April of this year. Feel free to contact me for more information about this particular workshop or my other work with mathematics and Scratch.

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- Making Math with Scratch
- MfA Workshop — Mathematics and Scratch
- MfA Workshop — An Introduction to Desmos
- MfA Workshop — Surfaces in Space

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When I see that the lights are off, I’ll step in, flip all the lights on, make eye contact with the proctor, and loudly say “The lights need to be on during testing”. Most of the time the proctor quickly averts their eyes, knowing they were in the wrong and embarrassed they’ve been called out. But one time a teacher, seemingly offended, responded “Well, no one’s complained.” I’ve heard a similar defense from teachers flouting school-wide homework and testing policies: “I give tests on whatever day I want. The students don’t complain. It’s fine.”

But there are lots of reasons a student might not complain when a teacher doesn’t follow the rules. A student may not want to publicly confess to poor eyesight in demanding that a teacher turn the lights on; a student who already has two tests on Friday may not want to risk upsetting classmates who would be happy if the teacher breaks the rules and gives them a quiz that day; a student may not want to risk possible retribution from a teacher by pointing out they aren’t following school policy when it comes to assigning homework.

Students exist on the wrong side of a perpetual imbalance of power in the classroom. Challenging authority is especially difficult under such circumstances, and in cases like this, students shouldn’t have to. We adopt rules and policies to protect student interests precisely because we know that young students aren’t always able to advocate for themselves. It shouldn’t be a student’s responsibility to make sure teachers follow the rules. It’s our responsibility, and our job, to follow them, even if we think no one will complain if we don’t.

]]>**Math**

*Genius at Play*, by Siobhan Roberts

I enjoyed this biography of the mathematician John H. Conway, which includes so many lengthy quotations that it often feels like a dialogue between the subject and the author. At times, it even feels like an autobiographical soliloquy, which I gather is not uncommon for Conway. The book covers lots of great math, too, which Roberts presents in an engaging and inviting way.

*How to Bake Pi*, by Eugenia Cheng

A fun, general-audience book exploring the author’s parallel passions for cooking and mathematics. And a gentle introduction to category theory, to boot!

**Teaching and Learning**

*Why Don’t Students Like School*, by Daniel Willingham

An overview of how some key results from cognitive science can inform effective teaching and learning. A book for practitioners, and a valuable read for teachers of any subject.

*The Teacher Wars*, by Dana Goldstein

A brief history of the teaching profession in the United States. I was surprised to learn that so many cornerstones of modern educational policy debate–tenure, training, curriculum, pay–have been argued about in much the same ways for over 100 years. An eye-opening read for teachers, and invaluable to those interested in framing current education policy in historical context.

**Non-Fiction**

*The Girl with Seven Names*, by Hyeonseo Lee

The fast-paced, harrowing tale of a young woman’s journey out of North Korea, full of twists, turns, and a good deal of sympathy for her homeland. An engaging and poignant look at a mysterious country through one person’s eyes.

*China’s Second Continent*, by Howard French

A journalist with deep connections to both lands explores the aggressive and strategic wave of Chinese immigration to Africa over the past 30 years. As someone who lived in China and wanted to learn more about modern Africa, it was a perfect fit for me.

**Science Fiction**

*Old Man’s War*, by John Scalzi

I ended up reading a lot of science fiction, and it started with this series from John Scalzi. Someone suggested that all the sci fi I read was a form of escapism, and it makes sense: these books were definitely a fun escape.

*Foundation*, by Isaac Asimov

Once I started with sci fi, I quickly gravitated here, and ended up reading quite a bit of Asimov in 2017. Having consumed so much sci fi in other forms (TV, movies), it’s fascinating going back to the source of so many of those themes, story lines, and even technical details. I was frequently taken aback at Asimov’s portrayal of the role of women in the distant future, which reminded me that stories about the future often tell us a lot about the present.

Thanks to these books, authors, and the Brooklyn Public Library, for making my 2017 healthier, happier, and better informed!

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