I can’t recall ever seeing this tiling before. It reminds me of the Old Woman / Young Woman optical illusion: Sometimes I see overlapping octagons and sometimes I see disjoint hexagons and squares. The photo also seems impossible to straighten: No matter how I rotate it, it always appears crooked in some way.

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So as a teacher I make sure students know that math will always create options for them. I tell them that whatever they decide to study — science, humanities, the arts — they should keep taking math classes as long as they enjoy them. There are quantitative aspects to every discipline, and knowing math will always set them apart and give them an edge in their field.

Recently a student asked me about how mathematics could be applied to the study of history. She is passionate about studying both, but sees them as disconnected and unrelated. I had a few answers for her, but I was looking to provide her with more. So I put out a request on Twitter.

I have a student who is interested in how mathematics can be applied to the study of history. I’d love to offer her a few entry points. Any suggestions? #math #mathchat

— Patrick Honner (@MrHonner) November 3, 2018

The response was remarkable. I learned a lot, and so did my student! Here is a brief summary of the great resources, links, and ideas that were offered.

- Mathematical Methods in Historical Chronology, an article in the
*Notices of the AMS* - Metric Geometry and Gerrymandering
- Quantitative History
- Network Models in History
- Looking at History Through Mathematics, book by Nicholas Rashevsky
- Social Sequence Analysis
- Optimal Matching
- Determining authorship of historical documents using statistical methods
- Dating procedures
- Applying Game Theory to the story of historical conflict (Cold War, etc)
- Building historical games (like
*Civilization*and*Age of Empires*)

There were many more responses, and I recommend looking through the Twitter thread. Thanks to everyone for contributing, and for helping to keep one more student studying math.

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Have you ever sat in a math classroom and wondered, “When will I ever use this?” You might have asked yourself this question when you first encountered “imaginary” numbers, and with good reason: What could be less practical than a number described as imaginary?

But imaginary numbers, and the complex numbers they help define, turn out to be incredibly useful. They have a far-reaching impact in physics, engineering, number theory and geometry. And they are the first step into a world of strange number systems, some of which are being proposed as models of the mysterious relationships underlying our physical world.

Some physicists currently believe that the octonions, an eight-dimensional number system with non-commutative, non-associative multiplication and seven square roots of -1, may be the key to understanding the fundamental interactions between particles and forces. Learn more about their connections to “imaginary” numbers in the full article, which is freely available here.

]]>We can also consider today a Transposition Day, as we need only a single transposition (an exchange of two numbers) to turn the year into the day and date.

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

]]>Here’s a peaceful scene from the West Coast, with a touch of mathematical appreciation.

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When I was really into guitar, I looked forward to my daily regimen of chromatic scales. I liked playing pentatonic triplets and G major chords up and down the neck, over and over again. Sometimes I’d block off an entire day just to practice. Part of me found satisfaction in the repetition, but I also knew that every note I played was another small step toward mastery. Sore fingertips and cramped hands were what learning felt like.

But there’s a lot more to learning than drilling. I knew it back then, on my way to channeling Mississippi John Hurt, and I know it now as a teacher. This repetitive practice worked as a part of a broader approach. Yes, I needed to drill. But I also needed to experiment, explore, collaborate, theorize, and reflect. Real learning requires all of this, and more.

So as a teacher it frustrates me when drilling is proposed as the remedy to society’s mathematical struggles, a topic discussed in this recent *New York Times* op-ed. Yes, students should know that 7 x 8 = 56. But they should also know how to think flexibly about numbers, so when they have to multiply 71 and 83, or 7x + 1 and 8x – 4, they won’t panic when they can’t find the answer in their times table.

Students should understand multiplication, not just perform it. They should know how multiplication can sometimes be thought of as repeated addition, and why sometimes it can’t. They should be able to interpret multiplication geometrically, as area or proportion. They should recognize the algebraic structures of multiplication in other mathematical contexts, like functions and transformations.

Facility with numbers and command of basic facts are a good start, but drilling can only take you so far. The note you want isn’t always in the scale you’re working on. To play along, you’ve got to learn how to improvise. To write your own songs, you’ve got to transform those scales into something fresh and exciting. This is what doing math can feel like. And this is what we should want more of for our students. Not more drills.

We know the dangers of pushing excessive practice. Ask anyone who gave up studying an instrument why they quit and the drudgery of drilling is likely to come up. Those who end up not liking math often tell a similar story.

And this emphasis on drilling can be especially harmful when partnered with out-of-touch portrayals of math instruction. Learning for understanding is not at odds with practice and fluency. Fun does not need to come at the expense of the struggle that learning demands. Getting the balance right for every student and every class isn’t easy, but that’s why teaching math is such a complex and exciting challenge.

Perpetuating these false dichotomies can actually reinforce the obstacles we face in learning math and improving education. They convey an inaccurate picture of what math is about. And they undermine the trust between teachers, parents, and students that success requires. Like excessive drilling, this can end up doing more harm than good.

*This essay was also published on the NSTA’s blog as part of my work as an NSTA / NCTM National STEM Teacher Ambassador.*

In my column, I use friendships to develop the basic concepts of networks and explore different structures.

When you start at a new school or job, or move to a new city, how do you go about making new friends? You could take an active approach, forging strategic connections with the popular kids and the movers and shakers. Or you could leave things to chance, relying on random groupings and associations. Whatever your approach, understanding the structure of existing friendships in your new community can help you make the best connections, which will ultimately define your circle of friends.

One particular network structure, the so-called *scale-free *network, has emerged as a useful model in a wide variety of fields. But recent research suggests that these scale-free networks may not be as ubiquitous as we might have thought. You can learn more by reading my column here.

The numbers of the month and day are a *derangement* of the year: that is, they are a permutation of the digits of the year in which no digit remains in its original place!Derangements pop up in some interesting places, and are connected to many rich mathematical ideas. The question “How many derangements of *n* objects are there?” is a fun and classic application of the principle of inclusion-exclusion. Derangements also figure in to some calculations of *e* and rook polynomials.

So enjoy Derangement Day! Today, it’s ok to be totally out of order.

]]>Happy rotational symmetry / double-reflective symmetry / palindrome day!

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The NYSMTP summer conference brings together Master Teachers from around New York state for two days of professional learning and networking. The theme of this year’s conference is “Convergence”, and features presentations from Dr. C. Alex Young from NASA, mathematician Steven Strogatz from Cornell, and New York’s Education Commissioner MaryEllen Elia.

I’ll be presenting *Scratch Across the Math Curriculum *with Dan Anderson, a Master Teacher from New York’s Central Region. Dan and I will be sharing our work bringing computer science into math class using the Scratch programming language. This is a continuation of the work I’ve been sharing at workshops and conferences across the country the past several years.

I’m proud to be a part of the NYS Master Teacher program through Math for America, the NYC-based organization that served as a model for the state program, and I’m grateful to MfA for their support in participating in events like the NYSMTP’s summer conference.

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