## Tuesday, March 13, 2018

### Walk the Line

I keep a close watch on this number lineI keep my eyes wide open for the signMaking sense, connections all the timeFor number sense, I walk the line

Apologies to Johnny Cash.

At Math In Action two weeks ago, I presented on number talks in the middle grades. (Here's the handout/resources, including a link to the slides.) I was quoting Pam Harris, then the keynote speaker was quoting Pam, so I made a comment on Twitter, badda bing, she turned out to be coming to my neighborhood the next week. Muskegon Regional Math Science Center let me crash, thanks Kristin Frang, so I got to crash for 1.5/2 days.

Pam was presenting on secondary number strings (thread 1 and thread 2 for my notes). Mini-review: I wouldn't hesitate to bring Pam in to work with teachers. Great energy, hilarious, solid ideas presented in a way that invites teacher access, and great modeling of instruction. All building on a great central message about giving learners an opportunity to mathematize.

I'm teaching Introduction to Mathematics this semester, a gen ed math class with a lot of freedom. I'm using Anna Weltman's This Is Not a Math Book as a text, mostly as a resource for learners and introduction activities.  In my head it's about redeeming mathematics for this successful students who have (mostly) learned to dislike mathematics. While we're mostly doing math and art, emphasizing problem posing and solving. But we take some time to redeem arithmetic, and we need to do algebra before we do patterning.

So, Pam's problem strings (modified because I can't help), followed by some clothesline math... rare day when I didn't have a way to take pictures. Sorry! The theme of the lesson is what else do we know?, which is one of my main understandings of math.

I drew a line, put on a hashmark, and labeled above it x, and below it 3. What else do we know? Some discussion about putting on a scale, vs knowing left or right. Finally someone shares -x is -3, and says it would go somewhere to the left. The someone said that means we know zero, which I encouraged as a good mathematician question.

First I did Pam's coordinates problem string.
(-2,5) show in graph, table, function notation
(2,-3)
(3,-9)
(1, __) which first brought up approximating with the line, then the idea of slope.
( __, 0) which brought a lot of people to a halt as they tried to remember an algorithm, then a remembered algorithm for solving equations. I made a bit of a joke how I was not interested in a memorized method, sorry, but only making sense.

Next, x is -2. Where is zero, left or right? What is x? Some discussion but pretty quick.

x-4 is 6. Is x left or right? What is x? How do you know? Quick discussion.

x+4 is -6. Heated discussion. -10 and -2 mentioned frequently. More and less start getting used more than left an d right. Interesting symmetry comparison between the last and this.

Then I introduced the clothesline, strung across the front of the room, with just an x in the middle. What else do we know? I had cards to ask them about. 0, -x, x+3, 5. We know where zero is, then dissuaded. What if it's negative? We know where -x is, then dissuaded. Then x+3, that has to be to the right. We don't know how far, but definitely right. What else? x-3 we would know. Excellent! Finally I put 0 up (left of x-3 by less than the distance to x) and ask what x could be?

Then I gave them cards. Make some cards, figure out the order you're going to reveal them and the questions you'll ask. Two really interesting situations came up.

One group had a couple of variable expressions, and then x+200. Really nicely subverting the sense of scale, and a great numeracy discussion of what x could be then. They wanted to just toss off a big number, but other learners argued for more precision.

Another group introduced a new variable, g, and then blew our minds with g^2. Has to be to the right, because squares are bigger. But what if it's negative? It's still to the right? Always? Then a lot of discussion if placement of it was setting the scale. Not until 0 is placed. It was amazing.

So that's my story. Thanks to Pam Harris, Chris Shore and these great learners.

## Saturday, March 10, 2018

### Let's Discuss Professional Development

One of my favorite math ed profs is Sam Otten at Missouri (and the Lois Knowles Faculty Fellow). His research is interesting and situated, he holds teachers in high regard and listens to their ideas, and he illuminates research through the Mathed podcast. He has definitely enriched my practice. In addition, he's just a lovely and creative guy, as well as a world class expert on the DC Comics film universe. Beyond that, he's a GVSU grad, so I knew him when.

He is a part of the team that produced some new professional development materials, and I had a few questions for him about it. Mathematics Discourse in Secondary Classrooms, MDISC, is based on research and developed with teachers in the field. I'm a big believer in the importance of discourse in learning, and know that secondary mathematics has been one the places where traditional teaching has included the least discourse. I also think people need support to make changes, so something like this project is needed.

What inspired these materials? Was it an idea you wanted to develop or a response to situations you saw in the classroom?
The MDISC materials came from a group of math education scholars at Michigan State University and the University of Delaware, led by Beth Herbel-Eisenmann, who were passionate about the role of discourse in math classrooms. We all believed that there was profound value in students discussing mathematical ideas and building meaning together as a community. So at its core, MDISC is a set of professional development materials that are intended to help teachers increase the quantity and quality of discourse in their classrooms.

As we set out to create these materials, we tried to draw on other work that already existed in the math ed literature. Some of that work was Beth's own research with Michelle Cirillo. They had worked for years with a group of secondary teachers, examining discourse patterns and power dynamics. We also drew on the work of Chapin, O'Connor, and Anderson, who wrote a great book called Classroom Discussions that focused on mathematical discourse at the elementary level. They had some really amazing results with respect to student achievement scores that stemmed from a new emphasis on discourse. With MDISC, we tried to take some of those ideas from the elementary level and reinterpret them in ways that made sense at the secondary level -- focusing on middle school and high school classes.

Overall, the MDISC PD materials equip secondary math teachers to think about discourse in productive ways and it also provides them with specific tools for changing the discourse in their classrooms so that it really empowers students. It helps move us beyond teaching-as-telling.

What are some of the different ways these materials might be used?
The MDISC materials include a physical facilitator's guide and then digital versions of all the participant materials as well as sample videos. It could be used by a teacher leader, facilitating sessions with secondary math teachers, or by a PLC of teachers who want to work through it on their own. It could also serve as a textbook for a graduate-level course, so a teacher educator going through the activities with practicing teachers, for example in a Master's course or an Ed Specialist course. The materials are designed to be a year-long study, with connections to everyday classroom practice, but it's flexible -- so with some adjustments, it could also be used in one semester. Or people could select which components they want to focus on.

There's also an optional follow-up where teachers can be guided through some action research, if they want to continue making purposeful efforts toward shaping their classroom discourse. There are several different options, and the MDISC team is very open to communicating with people if they have questions about enactment. We've also enacted the materials many times in many different settings, so we have a lot of experiences to share.

As you piloted these materials, what were some of the changes you saw in classroom discourse?
We have piloted the materials and had others pilot them in both Michigan and Delaware, with several different groups of teachers. They have been very well received thus far, with some teachers willingly joining in for a second and third year because once they start, they don't want to stop thinking about their classroom discourse. Some of the teachers have called it the most important learning experience in their teaching career, and this even came from a 30-year veteran.

The most visible changes have been the number of students talking in class. They open up more and share their ideas, and the great thing is that they're sharing mathematical ideas. I think this comes from MDISC's dual approach of not only providing insight into the nature of discourse but also providing specific moves for teachers to use. For example, MDISC develops six teacher discourse moves that include inviting student participation and also probing a student's thinking. These are concrete ways to get the discussion going and keep it directed toward important mathematics.

Another big change that is noticeable is that more wrong answers come to the surface -- it's not that MDISC leads to student confusion (just to be clear), it's that an increase in discourse helps more student ideas to come to the surface. And of course some of those ideas are incorrect or imprecise, and that can lead to good discussions and good learning opportunities for the group.

What’s one feature of these materials or an example experience that might help teachers understand how they will support their teaching?
One feature of the MDISC materials is that they are practice-based and case-based. So teachers will get to make constant connections to their own instructional practices and their own students. Those connections are built right into the materials. And there is also the chance to see and discuss detailed cases of other teachers. Rather than lots of little isolated examples, MDISC instead is built around larger cases of real teaching. So for example, when you're learning about the transition from small-group work to whole-class discussions, you can actually see a middle school teacher as she circulates among her students and selects certain ideas to be shared later, telling the students that she'd really like them to bring it up in front of the whole class. Then you can follow the case to see how it played out in the discussion.

Another important feature is that the MDISC materials integrate an emphasis on equity. Powerful discourse means that everyone has an opportunity to be heard and to learn from the conversations. So there is a lot of attention paid to how teachers can use a discourse-based approach to reach more students, including those with traditionally marginalized backgrounds.

What movie would you like to see DC make next?
Great question! When I'm not working in math ed or spending time with family, I love watching and analyzing DC superhero movies. I really loved Man of Steel and then I thought Batman v Superman took it up another notch, with great themes about immigrant experiences and the danger of overt masculinity having to face feelings of powerlessness. So although I'm excited about Aquaman and the Wonder Woman sequel, I would really like to see another Superman solo film make it onto the slate. And it would also be great if the Cyborg standalone would get the green light because I thought he was a really intriguing character in Justice League and I think his story could be used not only as a commentary about race in modern society but also about our increasing dependence on technology.

(Back to me) There's so much promising here. Use of real classroom discussions with connections to your own. The focus on equity. And the idea that in running it with teachers there's a measurable change in the number of kids participating in discussion, as well as the frequency and quality of discussions - that's a dream. I'd love a chance to work through this with teachers.

Find out more:

This has been a long time coming. Funville Adventures by A.O. (Sasha) Fradkin and A.B. Bishop is full of fun adventures.

Sasha is a Twitter acquaintance, an elementary math enrichment teacher with an amazing personal math journey, and I probably heard about the Kickstarter from there. I love to encourage these passion projects in general, but this book is especially delightful. (Sasha on Twitter & her blog.)

As a story, it may remind you in flavor of The Phantom Tollboth or Dragon Tales. Emmy and Leo are kids transported to an allegorical land, Funville. Kids in Funville each have a special ability. The story makes sense and is enjoyable without even knowing the math in a formal way, because the math is the idea behind the people they meet, but not how it's discussed.  These quirky characters are brought to life in quick vignettes and charming illustrations.

Part of the charm is that, since there are mathematical ideas behind the kids of Funville, the way they work and interact is surprising but logical.  Readers can predict what's going to happen or wonder what would happen.
“Yeah,” said Harvey as if it was the most natural thing in the world to have a power. “My power is to halve things in size.”
“Halve?” inquired Emmy. “But he is no more than a tenth of what he was!”
“That’s because ...
What do you think happened? Small mysteries like that. Big mysteries, like how will Leo get back to full size? Harvey has a brother Doug - will that relate? And the biggest: how will they get home?

Want to know more? There's a whole Funville Blog Tour, with lots of perspectives. As with a couple of those, you might find yourself wanting to make your own Funville characters. Even now, whom do you think Emmy and Leo might meet?

I have one here... Fan Funville Fiction!

Dylan's Dangling

Emmy and Leo worked hard all week to get done with school work and chores, so they would have an afternoon free to visit their friends in Funville. They had developed the habit of tucking things into an old rucksack that would be interesting to see just how their friends' powers would work on them. This time the rucksack held a tiny Ant Man action figure, an elephant toy, a stretchable rubber snake, and an assortment of snacks.

Emmy was particularly interested in Fay's and Randy's powers and how they interacted with other powers. So she was always glad to see them at the other end of the slide down the Thief. But she and Leo were both surprised to see someone new in the playground.

He was sitting on one half of a see-saw, but was up in the air instead of down on the ground. Maybe Heather had been here? He had on a shirt that was way too long, but otherwise seemed to fit him well. Adding to his stretched out appearance was a very tall cylinder of curly black hair.

Leo ran over to him immediately. "I'm Leo!" he announced, and the boy answered him. "Oh, I know. I came down here to meet you two because I was so interested in the stories that everyone told about you. People without powers, but you're fun anyway? And Pencilvania? Wherever that is!"

"Pennsylvania," Emmy corrected, "but that's a good synonym! You probably know I'm Emmy, but who are you?"

"Dylan," Dylan answered, "and I'm stuck up here. I was playing with- "

"Heather?" Emmy interrupted.

"Exactly!" said Dylan. "But her mother was calling, and she hopped off, and didn't notice that she made that end so heavy, and ... long story short, here I am."

"How can we help?" asked Leo.

"Do you have anything small enough to stand up under the other end?" wondered Dylan.

"Sure!" said Leo, and rummaged in the rucksack for a perfect skipping stone he was hoping Cory would be willing to use his power on. He slid it under the down side of the see saw. "Like this?"

"Mmmmmm hmmmmm," said Dylan, who was already concentrating. Slowly the see saw seat lifted up, pushed by the stone, which was growing. But not getting like a bigger stone - more like a tree. Once the see saw got level, Dylan hopped off.

"Not exactly..."

## Tuesday, January 30, 2018

### Book Club Fall 17

One of the many fun parts about teaching our capstone course, the Nature of Modern Mathematics, is the reading. Instead of me mandating a book (most instructors choose the excellent Journey Through Genius, by William Dunham), the learners can choose a book. Comes a day, we then have a book club class where people get to discuss with others reading their book, then share with the class what they thought. I try to keep notes, demonstrating poor steno skills.

The possibles this year are on this Google doc, which gets revised year to year.

How to Bake π, Eugenia Cheng

Connected all our classes, abstracted ideas but then super concrete accessible examples. Everything came together. Author is a little scatter brained: 15 subsections in each chapter. Even the toughest of concepts can be broken down. Two parts: what is math? What is category theory? Good connections.

Is God a Mathematician?,

History of math, Newton, Aristotle, Descartes… not proving that God is a mathematician, but looks at the beliefs of all these people. How does math intertwine with science, physics, biology… Example, knot theory. Is math discovered or invented?

Journey through Genius, Dunham
Miciah

Goes through theorem by theorem. Some was over my head, but the writer makes it very understandable. Example, quadrature of the lune. Most interesting was about Archimedes proof of the area of the circle.  Recommend it because it ties into a lot of things throughout our math classes, but you learn something.

The Teaching Gap

Compares German, Japanese and American lesson plans and how we teach. But mostly contrasting Japanese and American. In Japan they encourage more struggle. “US teachers are just not smart enough to teach the way researchers recommend.”

Joy of X, Steven Strogatz
Brian, Angel

Not especially challenging, written for a general audience. Longest chapter, 10 pages. Covers a lot of different areas of mathematics. Example, dating life. First half, playing the field, 2nd half find someone better than the first half… Snell’s law, ‘light behaves as if it was considering all possible paths … nature seems to know calculus.’ The focal points of the ellipse of Grand Central Station. Infinity. Is it odd or even? Recommend it. Even makes Hilbert’s Hotel understandable.

Fermat’s Enigma, Simon Singh
Proof of Fermat’s last theorem. Left so many conjectures, but the last one was a doozy. Made it as understandable as possible.

Genius at Play, Siobhan Roberts
Kelsey, Tony
More of a biography. He hasn’t published a lot, but his ideas are everywhere. He doesn’t like being known for the game of life. It’s hard to read, because the math problems are so hard. But you get to know his personality. See and say sequence from a student was frustrating, but then a source of great mathematics.

Quite Right, Norman Biggs
A history of time, … money. But 70% math. Start with caveman, then follow it forward. How to divide evenly, then follows through other cultures to modern math. Gives a sense of where math came from, but not all of it.

Finding Fibonacci, Keith Devlin

Story of Devlin finding the history of Leonardo of Pisa. Not recognized for his accomplishments. He didn’t really discover anything, but introduced real arithmetic and algebra. Son of a merchant. Really started a revolution. Only 14 copies of Liber Abaci in the world. Fibonacci sequence was just a puzzle in the book. Golden ratio, limit of the Fibonacci sequence. Does appear in nature, but not as much as people say.

e: the Story of a Number, Eli Maor
Most of the chapters don’t even mention e, but then it brings it back. Funny stories about many mathematicians (Bernoullis, Napier, …) Just a general  history, with some more focus on math. e is discovered, transcendental number…

Math Girls, Hiroshi Yuki
Math, but always in a story. Girls solve problems that have an easy access launch.  Someone who read this for a second book said it's mostly about the math content, but that content is deep and interesting.

The Man Who Loved Only Numbers
About Paul Erdös, an interesting, different, cool guy. Never owned many possessions, traveled from host to host working on mathematics. Took a lot of espresso and drugs, proved that he wasn’t an addict by stopping, but his math stopped, too. So he started back up. I really liked the prime number section.

I was able to entice a couple futute teachers to read José Vilson's This Is Not a Test for their 2nd book, and they were captivated, with strong recommendations.

...much time passes ...

I just now, in the next year, realize I never pushed send on this one! So >push<

### Chris Emdin #HipHopEd

Last night I got to hear Christopher Emdin in my own back yard. He was brought in to GVSU by the Black Student Union for Black History Month, without the College of Education or science educators even knowing about it. This is not going to be as much a recap as a response. I overtweeted during it as I think about that as my note taking now. (Here's the thread.) Saying he is a dynamic speaker is an understatement. He's the best presenter I've ever seen. It's a performance, it's heightened prose, it's preaching. Here's his SXSW keynote if you want a sample. And you want a sample. (Also his book, of course.)

So my response?

HELL YEAH.

This is my vision of education, expressed better than I ever could. It is about acceptance of all the varieties of giftedness and personhood and a chance for them to do deep, meaningful learning as themselves.

Dr. Emdin's emphasis on story telling as a way to share your own ratchetness and enter into your learners' world really resonates with me. David Coffey and I have been talking lately about just how can teachers share what they do. A Teach Off giving a lecture? No. Telling the story of what they do and why they do it and with whom they are doing it? Yes.

Part of making space for that story is accepting the pain of those rejected and making space for it and the healing. I love his idea of swag/cool/ratchet as the in-between of wound and healing. It makes sense to me and ties in to some pretty deep beliefs I have about redemption.

Chris warns against going into the hood (which can be anywhere that people are marginalized) armed only with the pedagogies of oppression. Dewey and Piaget and Vygotsky are still heroes to me, but that means we must contextualize them as well.

He offers no panacea, but inspiration. Progress is possible. Learning is local. And embrace your own ratchetness.

## 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.