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


(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.) 

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

Same Game, Different Grade

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:, 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.

Monday, May 22, 2017

Math Teachers at Play 108

I requested to host 108 because I have an absurd attachment to this number.

  • 108 beads on a Tibetan Buddhist Mala.
  • 108 pressure points in the human body.
  • 108 suitors courting Odysseus' wife Penelope.
  • An official Major League Baseball baseball has 108 stitches. In 2016 the Chicago Cubs won the World Series for the first time in 108 years, ending the longest championship drought in North American professional sports. The Cubs' win came in the 10th inning with 8 runs. From what perspective would you see the maximum fraction of stitches?
  • Police departments in India are warning iPhone users to ignore this Siri prank:  users say "108" to Siri, then close their eyes. When a user says "108" to Siri, it automatically dials local emergency services. (The prank ask users to close their eyes because there is a five-second window where a user can press cancel, in case a call was placed accidentally.)
  • Wikipedia notes: "The well known bas-relief carving at the famous Angkor Wat temple in Cambodia relates the Hindu story of a serpent being pulled back and forth by 108 gods and asuras (demons), 54 gods pulling one way, and 54 asuras pulling the other, to churn the ocean of milk in order to produce the elixir of immortality." Also, in karate, "The ultimate Gōjū-ryū kata, Suparinpei, literally translates to 108. Suparinpei is the Chinese pronunciation of the number 108, while gojūshi of Gojūshiho is the Japanese pronunciation of the number 54. The other Gōjū-ryū kata, Sanseru (meaning "36") and Seipai ("18") are factors of the number 108."

Late breaking news from biology via Paula Beardell Krieg: 108 ways to be crazy. 108/52 = 3^3/13...

  • The numbers 48151623, and 42 appeared in LOST, both in sequence and individually. The numbers add up to 108, and the button in the hatch had to be pushed every 108 minutes.

More mathily:
  • So why can we add 100 and 8 and get a number divisible by 9? That's weird, right?
  • It can be factored as 1x2x2x3x3x3. (Next up, 27648.) This means 108 is hyperfactorial
  • 108 degree angles in a regular pentagon, which makes it Golden ratio adjacent. Also 36 and 72 degrees are the angles in the golden triangle, which Euclid directs as the isosceles triangle with base angles double the other angle. 36, 72, 108...
  • I got wondering if many other numbers are multiples of the same number with (non-ending) zeroes removed. 
  • New to me was the idea of a refactorable number: a number divisible by the count of its divisors, also called the tau numbers. 108 is the 18th tau number. Can you find all 17 prior?
I discovered many of these things when (accidentally?) hosting Carnival of Maths 108

There weren't many submissions this month, so many of these are blogposts I've noticed in the past month. I encourage you to think about submitting - either your own or others - and hosting. Denise Gaskins is the originator of this carnival, and has a submission form. Last carnival at Give Me a Sine; next carnival at Math Mama Writes. (Hi, Sue!)

Long preamble! Let's Play Math, shall we?

Just, Wow!
Fawn Nguyen has many excellent posts, sites and presentations, but still occasionally has one that makes you say just wow. Her post on doing the Euclidean Algorithm with her middle school students was like that for me. Mike Lawler did an excellent follow up on the material with his family mathematicians.

Lana Pavlova explored what it means to be good at mathematics in this mindset-rich post.

Megan Schmidt blew me away with her rigorous honesty in her Ignite presentation on We Are Powerless. Link goes to the playlist, and you will enjoy every one you listen to. Only time for two more? José Luis Vilson, My Kids Can Do That Math, Too, and Robyn Drew, The Mystery of the Circle are where I'd start. 

Mike Lawler also got his boys exploring convex pentagon tilings via Evelyn Lamb's post on Math Under Her Feet. These are amazing geometric patterns.

Nice post about teacher's Math Circles by Katrina Schwartz at KQED/Mindshift. tl;dr: Doing math helps us empathize with students and teach better.

Chase Orton shared numberless data problems, extending Brian Bushart's numberless story problem idea. 

Denise Gaskins has a quick note about how even a workbook can be a moment for play.

Rupesh Gesota does an in-depth interview with a student on different division methods


Sasha (AO) Fradkin has a Kickstarter on for the book Funville Adventures, a math-inspired fantasy. (Her post to introduce it.)

Edmund Harris has some math-art shirts for sale. I ... can't decide.

Simon Gregg shared his students' responses to MANY Which-One-Doesn't-Belong situations.

Several of us enjoyed playing with the Girih Designer that Anugrah Andisetiawan‏ shared. (Islamic/PatternBlock tessellations and patterns.)

by coronaking
Paula Beardell Krieg completed an amazing book making and money math project with 2nd graders.

Watch Out 
Kyle Pearce shared his presentation Beauty and Complexity of Elementary Mathematics.

Kristin Gray has a whole sequence of number routines at the Teaching Channel.

Somehow I had missed the amazing Infinite Series YouTube channel from PBS, with Kelsey Houston-Edwards illuminating mind-blowing topics.

3 Blue 1 Brown released a terrific introduction to calculus in 10 essential YouTube videos.

Benjamin Leis wrote a response to Peter Liljedahl's Global Math presentation, and then tried out VNPS with one of my favorite mathgames, Sprouts. (This was a new-to-me blog I've subscribed to now.)

Advanced Topics
Mike Lawler also also got exploring complex roots & juggling with 3-D printing, sparked by a John Carlos Baez post. Beautiful, etherial structures.

Sam Shah did an intriguing exploration of Graham's Number. He's also looking 

I've invited the #MTBoS to follow along with my summer online calculus course. There are some cool teachers dropping knowledge on my students - awesome! 

One of the posts I'll share with those students is Simon Gregg's circle and square optimization post, with bonus generalization & striking images.

A couple of the images for this post were made with a GeoGebra factoring applet. How would you picture 108?

I'm getting to do a little (too little) in David Coffey's Math Recovery/Design Thinking course this summer. Here's his intro to SAFARI design.

Dan Meyer bids adieu to Malcolm Swan.  Dr. Swan moved the whole field forward and he'll be missed.

Wednesday, March 8, 2017

Lenten Teaching

Two things have me thinking about lent and teaching.

The first is just lent itself. An ancient word for spring, it's a fascinating spiritual practice; the idea of preparation for a holiday by engaging in disciplines. The traditional disciplines are prayer, fasting and almsgiving.

  • Prayer - most of my prayer time is in contemplative prayer. This can be thought of as emptying oneself, filling yourself with the Spirit, or relationship building with your higher power. For me it's a matter of prioritizing, because it's just as easy for me to put off prayer as it is to put off a phone call or email. (If I owe you either I apologize.)
  • Fasting - usually thought of as giving something up that you know you either particularly like or would be better off without. Lent is when my father realized he was an alcoholic, as he found himself literally unable to stop. (Later he gained sobriety through a twelve step program. for more than 30 years.)
  • Alms giving - often thought of as money for charity, more broadly it is service or caregiving for the marginalized, suffering or powerless.

For some reason, I've never thought before this year about what this personal life practice would be like in my professional life. I look forward to lent every year for it's sense of renewal. Maybe it's like New Year's resolutions, without having to pretend that you're going to be doing it forever.

What would these disciplines be like applied to my teaching life? Here's what I've got this year. (Hopefully I can do this every year as well.)

  • Prayer - on one hand, just praying for my students. I do this anyway, but have been more intentional about it this past week. Jesus knew what he was doing when he asked us to pray for our enemies. It increases compassion and empathy even there. How much more for people we already care about! But also, I'm trying to think about this in terms of relationships as well. What are the things I can do to strengthen my relationships with my students?
  • Fasting - this might be where I started. What do I do (or not do) in the classroom that I should give up? My goal is to try to interact more with individuals and groups while they're working. I tend to let them really work independently, and I don't want to start that, but it's okay, I think, to become a part of their group for a little while. Also goes with the relationship idea.
  • Alms giving - where can I be more supportive and generous to my students? All I can say so far, is that I'm on the lookout.

The second thing is a blogpost by Matt Larson, NCTM president, the Elusive Search for Balance. He gives some history about which of conceptual understanding, procedural fluency, and application have received more emphasis over the years, and advocates for balance, in alignment with the National Resource Council's 2001 recommendations:
"We want students to know how to solve problems (procedures), know why procedures work (conceptual understanding), and know when to use mathematics (problem solving and application) while building a positive mathematics identity and sense of agency."
The comments are also fascinating, with a lot of big math ed names.

My first reaction is that this is less of a pendulum swinging and more of a pendulum stuck to the procedural side of the triangle with chewing gum. When have we not emphasized procedure? I think in the research community we might swing a little, but in the teaching community emphasis on problem solving remains rare. NOT TO FAULT TEACHERS, as I have never known a community more focused on doing good for others. But we know that people tend to teach as they were taught, which does not push the pendulum.

But, of course, I do know a lot of wonderful teachers who are working in the balance that Matt is talking about. Thank you, #MTBoS. How did they get that way? We are drawn to systemic programs and sweeping curriculum changes, but that doesn't seem to change teachers.

What if it's more like discipleship? Teachers change when someone they know shares a better way with them. When their questions cause them to seek a solution and they find someone trying something that might help. It's not the person up at the front of the room with a microphone, it's the community of practice. This is something the #MTBoS gets right.

I've been thrilled with the increase in attention going to teachers like Dan, Fawn, Graham and Christopher. Their keynotes are amazing, and I've seen them light some fires in teachers' hearts. But we need to connect with those teachers and support them in this new direction. That is what's going to finally unstick the pendulum. Tell two friends and they tell two friends. Go hear Fawn together, then give a visual pattern or problem solving situation a try together, too.

To circle back to the twelve steps, the twelfth one is a doozy, and I think is what I'm trying to get at.
Having had a spiritual awakening as the result of these steps, we tried to carry this message to alcoholics, and to practice these principles in all our affairs.
Having found out what math learning can be like...

Of course, I'd love to know what you think, if you care to share.

Sunday, February 26, 2017

Math in Action 17

A highlight of February in these parts is Math in Action. Our local, 1 day math fest. Having been at the U for 20 years now, part of it is just great reunion, with our former students coming back to present and knock 'em dead. The last two years have felt stepped up, though, with a keynote from Christopher Danielson in 2016 and Tracy Zager, the math teacher I want to be, this year.

After taking a year off presenting last year, first ever, this year I was back at it to talk Math and Art with Heather Minnebo, the art teacher at a local charter that does arts integration. I've consulted with her, she's helped me a ton and we get to work together sometimes, too. (Like mobiles or shadow sculptures.) The focus this session was a terrific freedom quilt project Heather did with first graders. Links and resources here.

Next up for me was Malke Rosenfeld's Math in Your Feet session. Though I've been in several sessions with her before, I always learn something new about body scale mathematics. She ran a tight 1 hour session using Math in Your Feet as an intro to what she means by body scale math. One of my takeaways this time was how she made it clear how the math and dance vocabulary was a tool for problem solving. I often think about vocabulary in terms of precision, so the tool idea is something I have to think about more. Read the book! Join the FaceBook group!

On to Tracy's keynote. She was sharing about three concrete ways to work towards relational understanding. (From one of her top 5 articles, and one of mine, too.)

  1. Make room for relational thinking.
  2. Overgeneralzations are attempted connections.
  3. Multiple models and representations are your friends. 
Illustrated by awesome teacher stories and student thinking. She wrote her book from years of time with teachers and students looking for real mathematics doing, and it shows.  Read the book! Join the FaceBook group!

Plus, just one of the best people you could meet. She gave her keynote twice, and then led a follow up session. One of the hot tips from that was the amazing story of Clarence Stephens and the Pottsdam Miracle. 

 The only other session I got to was a trio of teachers, Jeff Schiller, Aaron Eling and Jean Baker, who have implemented all kinds of new ideas, collaboration routines, assessment and activities, inspired by Mathematical Mindsets. I was inspired by their willingness to change and by the dramatic affective change in their students. We had two student teachers there last semester, and it was a great opportunity for them as well.

Only downside of the day was all the cool folks I didn't get to hang with, including Zach Cresswell, Kevin Lawrence, Rusty Anderson, Kristin Frang, Tara Maynard... So much good happening here in west Michigan. Check out some of the other sessions and resources from the Storify

See you next year?