## Saturday, September 9, 2017

### What is Math?

It's the first question I pose for my capstone students, and then I ask them for the five biggest discoveries in math. A good way to start their blog (or reanimate if I've had them in class before.) Here's two of the previous classes responses: Winter 14 and Fall 15. The responses often run similar courses. Total aside: one student claimed firstmathblog.blogspot.com. How was that availabl

There's lots of math is everywhere and everything. I've felt and said that myself. One of my favorite teaching memories is a Kindergarten class that I visited weekly, and the first moment was someone getting to challenge me: there's no math in... bridges! Are you kidding?! Bridges are all about math! Next week, there's no math in Batman!

But is it helpful? If math is everything, then maybe people are already doing all the math they need to know. Me telling them that they're doing math either makes it irrelevant, or invalidates my line of reasoning because they very well do know what math is, they had a decade or more of it, and it is not that.

On one blog, I asked is the math the thing, the mathematical description of it, or the making of the mathematical description?

Lauren said that math is tool, but it's also an opportunity.  That's new to me, but also familiar. Isn't that the spirit behind #wcydwt and #anyqs? I had a little experience this week like that. I usually ask some kind of data question (mine or a learner's) on my sign in sheets and then make a display. Usually then shared on Twitter. For me it's a part of immersion, making the classroom mathematical. In some classes it leads directly into making representations, or becomes data for an activity. Almost always a chance to notice & wonder. Wednesday I shared one without the label, and it got some fun thinking. Chance for a joke, chance for some figuring.
 tweet

As for the milestones, I was struck by a few things this time.

• Numbers - lots of mention of numbers, sometimes specific like π, i or 0, sometimes familial, fractions or negatives. I am too eager to move past these, often, but now want to embrace them. The abstraction of quantity - that is a big freakin' deal.
• Pythagorean Theorem. Of course. But it is a big deal. I love its history, and continuing story, and think it must be one of the first examples of hey, this means this AND how can we use that? Thank goodness right angles are useful. Or are they useful because of this property?
• Patterns. So glad they think of this as essential. But when Eugenia Cheng says that math is the logical study of logical things, I think that math might have been born when we realized that there were patterns of patterns. When we were first meta.
• Euclid. One of the things that comes from the course is discovering the people in many cultures who took that step of writing and organizing what we know. There's something about math that makes it naturally becomes a system.
I love teaching this course, and learners who are ready to think about the meta-patterns are the main reason.

 Andertoons

PS>  I was listening to Anne Lamott's TED interview yesterday where she was so encouraging about just write. Just write. It really made me want to blog, to sit down and write. So when Lauren's post made me think that I wanted to think, I wrote. I can't worry about my blog being a bunch of first drafts. I can't be held back by the two open tabs on my Twitter Math Camp post and my summer calculus post. I just have to write. If you're reading this, thank you. That's already too kind.

PPS> If you don't watch the whole thing, you might watch around 25 min in (-15), where she talks about good writing is getting the reader to say "Ooh, tell me..." That set my teacher senses tingling. Her next part of that is that a confused reader is an antagonistic reader. That's exactly teaching, right? Where is the line between a learner wanting to know more, and not knowing enough to be interested. They need the beginning of a pattern, and to believe it's not just noise.

## Tuesday, September 5, 2017

### Top Ten Favorite Numbers

What numbers are the favorites of the people who have favorite numbers? I decided to ask on a lark, expecting a few responses, and it went crazy. (For my relatively quiet corner of social media.)

The idea had been bugging me since Joseph Nebus (who has a great weekly review of #mathcomics) linked to this comic from Cavna:

NO WAY are those the greatest, nor even the most popular. I can't even remember what tweet I saw that put this mild annoyance over the edge into asking out loud, but now I have a bunch of data on math teachers' favorite numbers.

This experience has taught me that our people care about numbers. They are more than quantities, they connect to ideas and stories.

Some things I noticed:

• 18 is the first natural number not to appear.
• Ironically, 2 is no one's 2nd favorite number, but is some people's 3rd or 1st favorite.
• Having a symbol or name makes you a Big Deal number.
• For about the top 20, number of mentions correlates to the Borda count (3 points for 1st, 2 for 2nd, 1 for 3rd).
• No one loves negative numbers. Come on people. Transfinite numbers got more love.
• 73 was the largest prime mentioned. Nope 163. Nope, 8675309. That number!
• 6 was the last single digit to be mentioned.
• 42 did not show up for the first several hours, then stormed up in popularity.

I'm going to show this list to learners and ask them to think about why some of these numbers might be on here. In particular the larger numbers...

My top three (not included in the data) would probably be 4, 0, and Φ. 4 was my first ever favorite number. I explained to several adults how it was both 2+2 and 2 x 2. As a joke I'd get people to continue the pattern 2, 4, ... and if they said 6 I'd say 8 and vice versa. Little pain in the neck I was. (Except I was never little, as the family joke went.) 0 is the competitive spot. 10 - the first number to show place value? -1 - the huge discovery or invention? Something with a slick math history, like $sqrt{2}$, e, 1729 or 163? Something exotic, like Graham's number, a googol, or τ? In the end I have to go with 0. The digit that became a number, with cool Bahmagupta connotations. Փinally, the number about which I sometimes tell students that it was invented by my great, great, great grandfather. Even if there was no name connection, even if it wasn't so marvelously algebraic, even if I hadn't seen 3rd graders discover it through the amazing Fibonacci connection, I would have to pick it for the spiral connections.

The question elicited some great stories and tweets...

I can be pretty dense.

Bob Lochel shared this perfect kickoff to the top ten, from the show that made Top Ten a thing. Also, when asked early in my career for what I wanted to be like, I often cited David Letterman. I apologize to my students then, and to their grandchildren.

So from the home office in Grand Haven, Michigan,

Math Teachers' Favorite Numbers

10.  It's the answer to Life, the Universe, and Everything...

9. Not really...

8. Moving up one space,

7.  Lucky for us, lucky for you, this prime is one better than perfect. It's been up, it's been heavenly, it's been deadly, it is in 7th place with 7 points,

6. Often considered the first number, and still the...

5.  Pythagoras may have called this number the root of all evil, and it still gets a lot of hype. It is geometric and irrational, ...

4. Move on, folks.

Nothing to see here. Except the number that makes our place value system so craaazy good; Brahmagupta made it work. Often mistaken for a vowel, sometimes seen wearing a fashionable sash. Er, slash.

3. Fee or Fie? You won't fo-fum when you contemplate this 3rd place number, unless you go to the point where you're crazy and see it every where. Favorite of the Egyptians, the Greeks and God if you believe all the hype, it's....

2. Popular choice among mathematicians, who have denoted it after the greatest ever to be called one of their number. It turns up everywhere, and has all your base.
1. As surprising as Alabama football, we find here the number with not one, but two days dedicated to it. Half the number some claim it should be, but twice what it takes to be right. A great big slice oooooof - no. I hate pie jokes. And what's with everybody focusing on irrational, when it's transcendental?

If you want to dig more deeply, Carolyn Frye recommended the great RadioLab show on favorite numbers.

If you want to math more deeply, here's the data in a Google sheet. Thanks to everyone who participated, and sorry for clogging up your twitter feed.

I think sometimes I protest too deeply the stereotype that math is all about numbers. Maybe there are times to just go with it, and geek out.

## Sunday, August 20, 2017

### Wand Shopping

Preface: Justin Aion makes wonderful wands. I made a joke about wanting to write a blogpost, then this story came to mind. Encouraged by Justin and Audrey McLaren, I decided to post it. If you have comments or suggestions, I'd gladly take them.

Ollivander's Wand Shop

Charlie (probably not the one you’re thinking of, if you were thinking of any particular Charlie) had her heart set on Ollivander’s, though her family had always gone to the Wand Mart. She didn’t have anything against her older siblings’ wands, and they did well enough in school. But she wanted her wand from the same shop Mr. Potter had gotten his. As did many, as well as the many who wouldn’t get a wand from where you know, er, Mr. Riddle had gotten his. Wand Mart had the latest Wizards Instruments wands, all cleared for OWLs testing at the factory. But Charlie had her heart set, and she usually followed that.

She knew her parents were sure they were right. Mom went to Hogwarts with a hand me down, and was very happy to send her siblings with new wands of their own. Of course, Dad had no idea before Alpha went to Hogwarts. That was a surprise owl!  Wand Mart reminded him of muggle school shopping, with most things in one place. And it worked out for A & B, so why would they mess with a successful formula?

Which almost explains why she was in Diagon Alley without her parents, who assumed she was in her room at the top of the stairs packing for Hogwarts. Headmaster Granger had sent quite thorough list, organized by category and even with packing suggestions. (She recommended moleskin bags, but they were not required. The “quite simple” charm had given her parents fits. Charlie’s sister Trish had fixed it up, muttering about the arithmantic beading patterns.) Charlie mastered the packing charm in no time, only slightly illegal for a wizard of her age, and that gave her time to sneak out. If her parents thought they heard rummaging sounds, Charlie had no idea the cause.

It was her first time in the Dalley alone, and she was excited. She resisted the pull of WWW, and scouted out Ollivander’s. When there was finally a moment with an empty shop, she darted in. Mr. Ollivander’s son Mr. Ollivander was startled. He kept glancing at the door, waiting for the ubiquitous & reliably following parent or parents. Charlie blurted “Mr. ollivander my parents want to go to wand mart but i know that the wands here are better why else would mr potter have come here for a wand twice even and i thought that if you well i was imagining your father but you’re just as good i’m sure helped me pick out a wand then i could just tell them see its all done and if there’s any difference i’ll pay for it myself because i think it really matters-”
 Handsome Kingwood wand, which Charlie didn't look at twice.  (Justin Aion's, really.)

“Um, Charlie. Carlotta if it bothers you to call me Charlie.”

“Why would it -”, Gerby thought and diverted, “So, you’re about to start Hogwart’s, Charlie?”

“Yes, sir.”

“I’ll tell you that many fine wizards started with a W.I. wand, and your parents have their reasons for picking that for you.” Charlie looked crushed. “But, if you want me to help you find a suitable wand, I’d be happy to help.” Sunshine returned to the shop.

“Yes! I need a Rosewood-Unicorn hair!”

“You do?! How do you know?”

“Rosewood is what Professor Sprout had and…”

“Mr. Longbottom, I suppose?”

“Yes! I’m a Gryffindor, too!”

“You know, I hope, that the wand chooses the wizard, but we can start there.” Gerby wandered out, muttered something, boxes fell, then “Ahah,” then “Bother,” then “Ahah” again. He returned with a reddish, short and rather thick wand with a plain handle, but intricate organic looking carving up the stem.

He held out the box, and Charlie oh so gingerly plucked it from the box. It felt… warm. Funny. Shaky. “Tell me what you think and/or feel,” Mr. Ollivander said.

“It’s funny.”

“Funny how?”

“Warm. Shaky.”

“Shaky or vibrating?”

Charlie thought about the difference. “Shaky. Do you want me to swish and flick?” She raised the wand as if to stop a marauding troll.

“No!” Gerby held up both hands. “Or, yes, but remember we’re in a wand shop. Lots of magic! Just gently, with your wrist” he demonstrated with his right hand “give a wave.” Though he didn’t have a wand in hand, Charlie thought she saw a fine mist of sparks trail his pointer finger.

Charlie complied. The wand felt as if she was pushing it. Sparks emerged, but big, with random direction and pacing. “May I?” Mr. Ollivander reached out for her hand. She went to put the wand in his hand but he said, “oh, no” and lifted his left hand to be under her right hand with the wand and held his right hand flat above hers. He leaned in and hummed, and the wand hummed back.

“The rosewood is fine, but I might like to try Japanese maple. It’s rare here, but would that bother you? But the unicorn hair is not the thing at all.” Charlie grimaced. “Sorry,” he said. “That doesn’t mean you’re not a Gryffindor, though I encourage you to be open about your house.”

He reached around under the counter, muttered what must be an inventory spell, and a foil covered wand box slid into his hand. When he pulled the box inside out, well, it wasn’t a lid, but more of a drawer, Charlie was confounded. “That’s not even a wand!” It was curved. Not like a bow, but, maybe a bit of a spiral? “How do you even know what direction the spell goes?!”

Gerby said, “I know. Unusual. Every wand is different, and I won’t make you try it. Maple, as I said, Japanese varietal, with the ruby red leaves, though the wand is quite blonde. It has a kirin scale edge core. Quite lucky, said the man who procured it but still had all his digits.”  Charlie set the rosewood wand in its box, and reached for the maple.

“Maple. Meh.” But when she touched it, it felt less warm than the rosewood. But comfortable. Picking it up - “ooh! This vibrates!” She swooshed it without being prompted, and it left no outward visual but instead made a sound. A fading single note?

“Ah, yes!” said Gerby. I had a good feeling about it. “Make a large circle, as perfectly round as you can.” She moved the wand as if drawing on a whiteboard, and made a shaky ellipse at best.

A disappointed, “oh.”

“Keep trying,” Mr. Ollivander said.

Charlie traced it again, and again, and again and soon the wand was drawing the circle by itself. Not exactly. Together? Somehow the circle just made sense. Mr. Ollivander rapped a knuckle in the center of the circle, and a gong sounded. Charlie giggled and Gerby laughed.

He leaned into Charlie. “The wand chooses the wizard, but the wizard learns the wand. How it is now, is good. But you will, or can, learn it. It’s a tool, which is how those W.I. … wands are made. But the understanding that works the tool is why it is your wand. I’m going to set this wand aside, and if you can convince your parents, fine. If not, know that it’s here waiting for you when you get a chance. Very nice to meet you, Charlie.”

“Thanks, Mr. Ollivander…” but he was already in the back of the shop. What an odd fellow. But - wow! What is a kirin? She was ready for whatever wand her parents picked, but she was already making the argument in her head why this was the wand. She wanted to reread Professor Granger’s list as well, with an eye for this learning idea. Somehow she had thought that Hogwarts would make her a wizard, but now she thought it might be more of her job.

Now what are the chances that Mom and Dad noticed her being gone?

## Sunday, August 13, 2017

### Sympathy Note

Trigger warning: learning styles will be mentioned.

I don't mean to mock trigger warnings or learning styles. I have colleagues who lose their mind at the mention of learning styles, because of the lack of research. And we've all seen people flip their lid at trigger warnings. (How do you warn them?) I like to think about learning styles as a framework that teachers use to make sense of what they see. And to think about what they might do in response.

I love to read, but don't have as much time for fun reading as I like. One benefit of driving to Twitter Math Camp (about 12 hours each way) was the chance to listen to a few books. I mostly read mysteries and science fiction or fantasy. On the way home, I listened to a Hieronymous Bosch mystery by Michael Connelly, usually quite good. This one was read by the actor who plays Bosch in the streaming series (thumbs up for that, too), so it was an interesting experience. I was using the Libby app, which connects to public libraries and was overall great. Through Libby you can search catalogs and make requests, and I requested the next book.

Turns out, I requested an audiobook. I don't have time to listen to a book... but, I had it, Titus Welliver (the actor) ... okay. I'm listening to it. But I have to be doing something else at the same time. I've never been able to just sit and listen. In school I was a doodler. For which I, and then later my kids, received plenty of disapproval from teachers. (Until grad school, when profs wanted copies of my notes!)

Reading a text, I have great recall and comprehension for names, plot, etc. I notice small details. I concoct theories. Listening... I'm still enjoying it. But will suddenly be 'wait, what?' The interface is handy, but it's been hard to back up. Whereas in a text, it's easy. I'm not following as well. Reading I can visualize the action like my own movie. Listening... not so much. "Wait, what?"

Is it a learning style? Am I a visual learner? Textual? Is that a thing? Inexperience vs experience? I don't know.

What I do know, is I'm committed to is remembering to say and write important points for my students. For having learning opportunities in a variety of modes, including motion.

Now I've got to get back. The bad guys have Bosch and the doctor cornered in the doc's office after hours.

## Monday, July 31, 2017

### #ITeachMathLearners

(Have to read that post title Sixth Sense style.)

First do I write about: #iteachmath or Twitter Math Camp 17? ...  have to get the hashtag stuff off my chest.

I love that Dan is thinking about inclusivity, and it befits his problem solving orientation that he's willing to rethink any aspect of the situation.

I started blogging April 2009 with a 50 word post, just sharing a resource I liked. I thought I would use the blog to share the stuff I found around the web that I like or was thinking about how to use in my class. Ten posts later I finally shared an first activity that I did back then. A math game, of course.

This was a long time ago in internet years, and I understand that the world is different. I was inspired by what I was finding, and just wanted to join in.

First tweet, 2010.  (Find yours.)

I was at Maria Anderson's tech camp (@busynessgirl) and she suggested Twitter as a way to connect with student teachers. That's been great, but I wound up liking the math twitter/community plenty for myself.

When it was time, 2013, the community wondered how to refer to itself. They came up with Math-Twitter-Blog-o-Sphere, and I liked the silliness of it right off.

Is #MTBoS a barrier now? People are hurt by this suggestion, because they work like hell to make the community inviting and inclusive. And are always looking for more ways to do that better.

From where do the hard feelings come, then? I have theories. Basically this list is the consequence of people new to twitter don't know how it works yet.

• some of the most followed people are friends. They take math with anyone, but also talk real life to each other. There's shared experiences, so they refer to things that not everyone was a part of. But because of the way Twitter is, we see some of those relationships. That could make you feel like (Justin Aion analogy) being at a party by yourself. As Justin says, at a party, they'd see you standing alone and approach you, but on Twitter, you can be invisible if you want.
Remedy: new users can let people know they are there.
• People say 'Include #MTBoS and get your questions answered.' Sometimes? More people watch that tag and respond to new people than I think would ever happen in most communities, but not everyone gets responded to.
Remedy: tweet @someone. If I see someone asking for a resource, I may not have a response, or know others who have better. But if you tweet @mathhombre, I reply. (I think?) I challenge you to find a community with a higher response rate.
• #MTBoS is a community. We have relationships, shared values, and even meet when we can. If there's an in, there's an out.
Remedy: come on in!
What I notice about these problems is that the remedies are all putting the burden on the people who feel outside, which is usually the hallmark of an exclusion problem. But that's where we need to see and popularize the efforts of Tina Cardone, Sam Shah, Lisa Henry et al. There wasn't an intention to brand anything, but having a name is part of making you a group, a tribe or a family. I would rather reassure people that this family wants you and is inviting you in, than worry about what the name connotes.

It did feel autocratic, and like a dictate, but that's probably mostly because of his position in the community. He is the introduction to the MTBoS for most math teachers. He is going to hear the complaints the most, maybe?

The timing was really unfortunate, as it distracted from the amazing keynote by Grace Chen. (Pts 1 and 2) (Which I will talk more about in the next post.)
 Andertoons

Dan is trying to connect people with #iteachmath. Great! I don't see how that solves any of the three bullet points above. Hopefully, the new tag will be successful. If it is, within a couple years people will feel like it's cliquish or there are rockstars and arguing for #mathlearnersunite. Great!

I don't think of this as particularly important - or coherent - post, but this is a blog. I can work out my thinking here, and live long enough to be embarrassed of it. I can give a first take. No one else may read it or maybe it becomes the rare post that gets a comment. One of my Twitter Math Camp take aways, from Carl Oliver's sweet keynote (Pts 1, 2 and 3), is that it's important to push send.

If you hear about the #MTBoS, my guess is that you will be curious enough to investigate. If you do investigate, you'll find things that will help your learners. If you value that, I encourage you to join in. The more you participate, the more you'll feel a part. If you can get to a Twitter Math Camp, you'll be stunned at the welcome. But nobody's going to make you.

## Monday, July 24, 2017

I love working with people. Given the choice of math together or apart, or teaching together or apart, I would pretty much always take together.

So I was thrilled when Joe Schwartz was willing to work with me for Twitter Math Camp this year. Among many shared math interests, we both love math games. I've learned a lot from Joe's math game posts, and his Twitter Math Camp 16 presentation on them blew my mind. Such great learning potential the way he approaches games with students.

 E.H. Shephard illustrating A.A. Milne
One game with which we had both already done things was Fill the Stairs, which has been around for a while in many variations. I had done a version with Esther Billings and David Coffey in an inservice, that involved having cut up bags of numbers. Joe had done a version with digit cards 0 to 9 where students fill in a staircase with 10 on the bottom stair
and 100 on the top stair. Joe and I had connected on a version I came up with earlier called Decimal Pickle. As we talked about it, the spirit of Tracy Zager began again. She's been haunting me all year from her TMC16 keynote, where she challenged us to do cross-grade collaboration. "What if we did variations of the game across grades?" Joe wondered. So we were in.

Necessary references: Joe's first post about it, and the redux.

The idea for Decimal Pickle came from a need for comparing decimal numbers of different length. Why would we flip different numbers of cards? The colors are pretty intuitive there. Black? Flip again! It added a lot of excitement to the game, almost a black jack feel. For the mathematics, it was perfect for the 5th graders to compare tenths, hundredths and thousandths.

Talking with Joe got me thinking about the big topic. How does order show up across the grades? When I think about number sense, I see a few components. First, number as quantity. Or the numbers in context. But second only to that is comparison. Well, second is representation. But third only to those two is comparison. And comparison before computation, which is right out. But this idea of order is really an up and down the curriculum issue. As numbers grow more complex, how to order them is very relevant. It's the experiential aspect of number that we often ignore as we get farther up the curriculum. One of the strengths of this game is that it requires comparing more than two numbers. I think ordering a set is more complex and challenging task. There may be a component of number sense I haven't thought about at play, a kind of sense of distribution.

I like games to use easily accessible materials, so playing cards are great. I often use J as 0. (And if the kids are old enough, "You know what you have if you've got Jack?")  I'm not sure why I first tried having students make their own gameboard, but I love it, now. There are students for whom that's their in for the game. (Deep game design - there's probably a whole player psychographic aspect to this.)

The first step for me on this generalize the game journey was to fit it for 3rd graders. I wanted single digit and 2 digit (teachers asked for no three digit), but thought that half and half was a weird balance. I settled on turn over a card. On a diamond you stop, otherwise turn over another.

Kids are not as familiar with playing cards as they used to be, so we started with something halfway between notice and wonder and Which One Doesn't Belong.  Then, as I often introduce games, I played vs the entire class. Then they break up and play in 2 vs 2 teams. One of Joe's great ideas was to have students make a number line with their results. Great task, ripe for discussion, strong in representation, awesome assessment.

Thinking about how to go even younger, I was thinking about sorting single digit cards. But how to make a game out of it? First came the first grade variation. Fill five spaces. Flip cards like War to start in the the middle space. Higher goes first (advantage) but their card is probably too big for the middle space (disadvantage). Every flip you place in a spot. If it's the same as a card you have, cover that card. Cards have to stay in order. So if you have 3 __ 4  6  __   __ and you turn over a 5, you can cover the 4 or the 6, but not the blank between 3 & 4.

That requirement to not move cards was too much for Kindergarten, so they could move their cards around. That generated plenty of discussion, too.

Thinking about how to extend the game past high school was a challenge. I kept thinking about order of operations. One of my pet peeves is PEMDAS, as I want students to think about 4 levels, grouping - exponents - multiplication/division - addiition/subtraction. (GEMS if we need an acronym.) So I thought about doing the red/black for more cards. 2 cards minimum, you can add or subtract. Three cards, have to join with add/subtract and multiply/divide. Four cards, have to do an exponent or root. Five cards you have to use a grouping structure, parentheses, radical, fraction bar. I think this could work, but haven't got a chance to test it yet. The class in which I was going to get to test was a college algebra class working on exponents, so this became the variation.

 Marcel Duchamp
It was great. The students were constantly surprised by their results, got a lot out of comparing these extreme numbers, and became more efficient at arranging the numbers to get the effect they wanted even in the course of an hour playing the game. Older students also need these play experiences, I think they just abstract from them more quickly than younger students.

We'll be talking about this at Twitter Math Camp 17, so I hope you can join us if you're there. Here's a page with downloads and resources: http://bit.ly/stairs-tmc17, regardless. If you have ideas for more variations or get to try one of these, let me know!

What games do you use that connect to a big idea in math?

## Saturday, June 17, 2017

### World Tessellation Day 2017

Happy Tiling!

Our 2nd annual World Tessellation Day, celebrating the birth of Maurits Cornelus Escher on June 17, 1898.

Pat Bellew has a good quotation in today's On This Day in Math:
By keenly confronting the enigmas that surround us, and by considering and analyzing the observations that I have made, I ended up in the domain of mathematics, Although I am absolutely without training in the exact sciences, I often seem to have more in common with mathematicians than with my fellow artists. - M. C. Escher, Quoted in To Infinity and Beyond, E Maor (Princeton 1991)
The best post I saw in advance was Evelyn Lamb's Look Down. (Which also sounds like it could make a good horror movie.) Also note the founder of this here holiday's post, Emily Grosvenor's post on making the holiday. Also also note that Eric Broug's  TED-Ed on Islamic Tessellations came out in time for today.

Best way to see what is happening is probably the twitter hashtag #worldtessellationday. Lots of groovy there.

I had to make something, so here's a GeoGebra applet. Instead of an Escher-style, I made a square you can fill and then fill the plane in three different ways. The matching conditions make for some cool patterns to me, like infinity tiles. If you want an Escher style applet, check my collection of tessellation sketches.

My most recent tessellation work was with Heather Minnebo's art students. I helped some work out the mathematics, but got interested in her art directions.

I was having to eavesdrop while taking to other students, and we'll talk more about. But what I got was her talking about size of the tile compared to the final paper (encouraged to be quite large, like a meter by 1.5 m.) She talked about the designs within the tile, that were going to have to fill the whole paper. It struck me that the same things she was looking for are what I would want to emphasize the mathematical structure. Plus some nice visual estimation. Afterwards, she was highlighting craftsmanship as a growth area for some students, and I was wondering what that looks like in the math classroom.  Heather says "craftsmanship for my kids (excluding some of my quirky friends who are owned by their ideas) comes down to ownership and then honing skills." She notes that at least half-ish of them were able to mesh conceptual understanding with some solid technical skills and creativity.

Heather notes: I have more questions than observations.  I've taught this lesson now half a dozen times, how long before I really get a handle on how to best teach it? Every time I get an idea for how to improve the instruction and pacing, but it's a long ways from solid.  (A solid chunk of this weakness/uncertainty is the knowledge that I don't have the full breadth of understanding of the mathematics behind it.)  How do I get them to keep the spontaneity of their creativity and experimentation, but add in the understanding that this game has rules to follow. For  instance, when I say midpoint I mean get a ruler and find the midpoint.
The enthusiasm and interest is palpable from the introduction – their minds are blown by the Islamic architecture and Escher's work – to their own initial trial and error.  How do I encourage that same level of interest and enthusiasm all the way through to the end?  I have two groups of kids who are able to do this from beginning to end: the ones who get it, own it, and have the discipline and mastery of skills to carry out their vision, and then there are my creative geniuses who are owned by their ideas and they are consumed with fulfilling the vision in their heads and I don't believe they see the issues of craftsmanship or skill we see.  The bulk of the kids are in the middle – their enthusiasm and effort peters out (in varying degrees) as the repetition of shapes try their need for instant gratification, brains.  This harsh judgment includes myself as I too often fall into this  low attention span category.
Lastly and most complex, how do I convey the challenge to see each individual shape as a separate defined image/area, yet also view and plan them work together as a whole?  Unbelievably, More than a few kids ignore the pattern they've established by tracing their shape, and they add color and additional patterns over all – sometimes obliterating the tessellating pattern they created.  This blows my mind.  I struggle against being frustrated as I think they absolutely must be missing something to choose to do that.
Those are just a few challenges and questions I have… I'm sure there are many more this was just off the top of my head.

Wow. These are some of the biggest teaching questions that there are! Any suggestions or comment from readers?

I always love how bold elementary students are (compared with college) and willing to try when college students often need to know if it will work first.