PD Like It Oughta Be

    It started with an e-mail, and ended as a powerful professional development experience.  Along with five other members of the MTBoS...

....I was invited by Andrew Stadel to participate in the trial run of a new 3-Act task he had developed. I would be required to collaborate with a partner, and despite the fear of having my math inadequacies exposed, I put my faith in Andrew and agreed.  (You can do the 3-Act too!  Just follow the above link.)
    We started off working alone.  He sent us the Act 1 video to watch, and asked us to respond to the following prompts:

• What do you notice?  What do you wonder?

I shared my responses in a google doc, and it was collected there along with all the others:

• You're the only one on the playground. You pushed that swing really hard, but not hard enough to wind it all the way around the bar.
• Will the swing make it all the way around this time, or will you need to push it again? (I'm thinking it will). Is that as hard as you can push? I'm wondering how long the chain is and how high the cross bar is off the ground.

     Andrew commented on our responses, and then popped the question:
     How many total times will the swing wrap around the pole?

     First we had to make our estimates, and provide some reasoning.  My too low was 2 wraps, my too high was 10 wraps, and my "just right" was 6 wraps.  Why?  I had watched and re-watched the video, and it looked to me like 1/3 of the chain was wrapped and about 2/3 was left to go.
     Following 3-Act protocol, we needed to request the information we felt we needed in order to be able to figure it out.  I wanted to know two things: How long is the chain?  How much length does each wrap take off the chain?  Again, our responses were collected in a google doc.
     Andrew's next e-mail contained the information for Act 2:
   

     We were asked to come up with a representation for our solution, then set up a Google Hangout and share with our partner.  Mine was Shauna Hedgepeth, known in MTBoS circles as Hedge.  I found out through her blog that she is a K-12 math coach in Mississippi with classroom experience teaching 7th grade, Algebra 1 and 2, and AP Statistics.  I knew I would have to up my game.  We set a date and time, and the clock started to tick.

My first instinct was to draw a picture.  After all, isn't this what I always tell the kids to do?  On my first attempt at a solution, I rounded off the chain length to 7 feet and divided that by 2 1/2" , the diameter of the pole.  (The diameter was actually 2 7/16" , but again I rounded).  I got an answer of 2.3 wraps, which I knew had to be wrong.  Why?  Because it made no sense!  The swing had already wrapped around twice and there was still lots more chain to go.

   A day or so after this first, failed attempt, I realized that I needed to convert the chain length into inches before doing my division.  I felt pretty good about having figured that out, but when I tried it (see second attempt), I got an answer of 33 wraps, which again I knew had to be wrong!  How did I know?  Because of my estimate.  33 wraps was way too many.  It wasn't reasonable.  It was time for another approach.

A search around the house turned up a poster tube.   I found a chain in the garage. 

I wrapped the chain around the tube over and over again.

       It didn't help.  No matter how many times I worked with the model, I could not figure out what I was doing wrong.  I would have to confess to Hedge that I could not come up with an answer that made sense.  But I knew enough about the MTBoS community to know that it would be a safe place for me to fail.  I was confident that the faith I put in both Andrew and Hedge was not misplaced.
   The day before our scheduled Hangout, I happened to glance down at the tube.  I saw it from a different perspective:

I stared at it for a few moments.

    Suddenly I felt something click.  "The chain isn't wrapping around the diameter of the pole, it's wrapping around the circumference!  You need to divide the chain length by the circumference, not the diameter!"

After a quick search for the circumference formula, I came up with an answer of 11 wraps.   I will say here that, except for the very basic computation, I did all my work with my cell phone's calculator.  Remind me again why I'm spending so much time teaching kids how to multiply and divide decimals like this with paper and pencil?

    11 wraps was more like it.  My intuition told me it was reasonable, but it was still higher than my estimate of 6 wraps.  Was there something I was missing?  I went back to my model.  My chain was 28 inches long, and the circumference of the poster tube was 7 inches.  Using my formula of dividing the chain length by the circumference,  the chain should wrap around the tube 4 times.  However when I tried it, the chain wrapped around only 3 times.  It was easy to see that the chain's metal links prevented it from laying flat against the tube.  I would have to take that into consideration.  But how?

I set up a ratio.  My thinking went: If a chain that should wrap around 4 times actually wraps around 3 times, how many times will a chain that should wrap around 11 times actually wrap?  8.25 times!

   Closer to my estimate of 6!  I felt good about my work, and looked forward to meeting Hedge.  I was a little nervous, but we hit it off right away; she thought I might make fun of her accent, and I thought she might ask me, "What exit?"  We spent time sharing our work and our solution strategies, and as it turned out she came up with an answer close to mine.  She had drawn a diagram too, and I admired the way she went right to the formula.  She liked my use of the poster tube and the chain.
   The time flew by.  The fact that the child who had grown up and lived for so long with such a tortured relationship with math had just spent a half hour talking with a math teacher in Mississippi about how to solve a math problem was nothing short of amazing.  It was a testament to the redeeming power of the MTBoS.
    For me, Act 3 was anti-climactic.  The fact that the swing wrapped around 7 times and not 8 didn't bother me in the least. I felt successful.  The final debriefing took place in a Hangout that spanned 5 time zones and the entire country, from New Jersey to Mississippi to California to Hawaii as Hedge, Sadie, and I shared our experience with Andrew.  For myself, I had 3 big take-aways:

• The estimate I made helped me discover my mathematical errors.  I knew that my first two answers could not be correct.  They were not reasonable given what I had seen in the Act 1 video.  I needed no math to figure that out; only my intuition.  This reinforces my belief in the power of building estimation skills through estimation180 type tasks.  The big move will be to get students to generalize that kind of thinking to all their problem solving experiences.
• The anxiety I felt about having to share my work with someone I did not know, and the fear that I could be totally wrong and might embarrass myself, was alleviated by the fact that I was sure I was working within a community that was safe.  This reinforces the importance of building safe communities within our classrooms, where there is a culture of collaboration and the fear of failure is banished.
• Most importantly, Andrew gave me the gift of time.  I'm sure that the others came to a solution quickly, but I needed days to process this task.  I needed pictures, physical models, time to fail and fail again.  The "a-ha" diameter vs. circumference moment was an exciting moment of discovery, a moment that would have been robbed from me had I been working under a tight deadline.  Unfortunately time is a luxury we do not often have in school.  It made me wonder: How many exciting, "a-ha" moments are robbed from students simply because they do not have time enough to think things through?  

     It was an enriching project, one that was important for me to experience.   As a wise man once said,
     "You never really understand a person until you consider things from his point of view...Until you climb inside his skin and walk around in it."  I'm so used to being in "teacher mode" that it's easy for me to forget what it's like to be a student.  This was a powerful reminder.  Another gift from Andrew.
     Weeks later, a third gift came in the mail:

My very own Estimation 180 t-shirt.  Thanks Andrew!

The Bestest Things in the World

    One of the things I love about being an elementary school math specialist is getting to work in primary grade classrooms.   They have lots of cool stuff down there!

Links!

    With Mr. Harris as my inspiration...

4+4=8 known fact. Knew 6 was even and so is 8. Counted up by 2s from 8. This sculpture LOOKS like her strategy! pic.twitter.com/uNLkJ1rblh
— Mr. Harris (@Mr_Harris_Math) September 19, 2014

... I thought I might try them with the first graders to explore visual representations of complements of 10.  I am fortunate to work with some amazing first grade teachers who allowed me to experiment.
     First we let the kids play around:

They went wild.  "These are the bestest things in the world," exclaimed one girl.  A boy remarked, "This is better than having an i-pod!"  They made necklaces, bracelets, key chains, swings, snakes, and handcuffs.  They linked them together in patterns, measured them, and sorted.  All without any teacher direction.

After about ten minutes, I called them all together.  I showed them a chain that I had made and asked:

    I asked them to turn and talk to a partner, and when I called on a student, I asked him to share what his partner noticed.  They had quite a bit to say, including observations about the colors, the total number of links (10), and their arrangement.  Some children saw 2 red, 6 yellow, and 2 red, others 4 red and 6 yellow.  Others saw 5 and 5: 2 red and 3 yellow and 2 red and 3 yellow.   I explained that they were now going to make chains of 10 links, but that they could use no more than 2 colors.

This was a popular arrangement: alternating colors 5 and 5.

Everyone was able to make at least two chains.  I added another stipulation after reviewing the results from the first class: once you had used a combination (5 of one color and 5 of another, for example), you could not repeat.  This was because, out of 44 chains, 35 were 5 and 5 combinations, 5 were 10 and 0, and 4 were 6 and 4.   I wanted representations of all the complements of 10.

The next step was to record the chain on a piece of paper.  The first class drew only the links, while the second class recorded the number of each color used.

     This took about an hour.  I collected their work and came back several days later with...

...strips of paper to make paper chains!

The kids had to look at their original picture and create a paper chain.  We asked them to attach an index card with a number model describing how they made 10.

The completed chains made for a colorful display, and a reminder of the complements of 10.

Turns out the project had many benefits:

• Linking and gluing promoted fine motor skill development.
• Students had practice sorting as well as creating patterns.
• The process of turning their pictorial representations of the chains they had created with the plastic links into paper chains required an attention to detail.

  Now that the students have the procedure down, they can begin to compose and decompose other sums.  I know there are more links hiding in closets and collecting dust on shelves.  Let's put them to use!

The Math Message is Dead! Long Live the Math Message!

     I see this in many classes I visit:

It's a Math Message!

     Back when I was in teacher school we called this a "sponge activity".  You might call it a "do now". Either way, it's meant to be something kids can work on for a few minutes as they transition into math class.  Teachers have it up on the board, and the kids copy it down in their math notebooks. Maybe it's a skill review, maybe the introduction to a lesson, maybe something to keep the kids occupied while the teacher gathers his thoughts and plans together.  Whatever it is, it has to go away.

Here's an example from the 5th grade manual.  Typical scenario: 4 kids are finished in 5 seconds, 5 kids can't find a pencil, 3  have no notebook, 7 are trying to multiply 37 x  62,  and the rest are praying that the teacher won't call on them. 

Here's one from grade 3.  For classroom scenario, see above.

Another gem from grade 3.  This question may have been interesting to its writers, but I have seen it cause many 9 year-old eyes to glaze over.

Here's a sample from grade 2.  How long do you think it will take a second grader to get all that money out of his tool-kit?  Paging Mr. Stadel.

Finally one from grade 4.  How long should the teacher wait for everyone in the class to be prepared to read the numbers aloud?  

     I confess to having used these on a regular basis when I was a classroom teacher.  But stepping back now as an observer, I have come to the conclusion that we must re-imagine what a Math Message can and should be.
     Last year the fourth grade teachers agreed to scrap the traditional Math Message in favor of daily estimation180 tasks, something I've blogged about repeatedly (here, here, and here for example).  Theresa and I encouraged the third grade teachers to replace their traditional Math Message once a week with estimation180 tasks, some of which were created and produced by the fourth graders.  We even got the fifth graders into the act, with tasks from the estimation180 site along with ones Theresa and I created to tie into the curriculum, including a series on volume:



volume1movie from Joe Schwartz on Vimeo.


and another starring a balance scale:





1marker from Joe Schwartz on Vimeo.


     While we were happy with the results, we felt that this year we needed to expand the repertoire.  So we've started to experiment with two new "do now" tasks in the hopes that we can add them into the Math Message rotation.  Neither are revolutionary, but we feel they represent a vast improvement over what they're replacing.
    The first is "Always, Sometimes, Never".  Here's one Rich and I tried out:

Seeing this up on the board for the kids to mull over made me happy.

     The kids got to work in their notebooks, and while they may not be so great at explaining, defending, or proving their positions yet, we do see levels of engagement, thinking, and excitement that we rarely see during the traditional Math Message.
    Many students felt it was "always", though several students did mention decimals, one brought up negative numbers, and, "Is 003 a 3-digit number?" was a question that got lots of debate.  One student even tried to rewrite the statement to make it Always True.  It was a mathematically rich and engaging 5-7 minutes, it reinforced concepts we're teaching this unit (and that we revisited during our mid-workshop interruption when we looked at the problem 8.4 - 5.73), and completely blew the doors off finding out how many days older Amy is than Bob.
     The second is to ask the kids to do some noticing and wondering.   I tried this out in second grade, when I used the technique to introduce a game.  I've encouraged teachers to use 101qs as a source of interesting pictures that can inspire kids to notice and wonder.  And it wouldn't require much work to turn this:

Into this:

  What conversations might this Math Message provoke?  What type of thinking might it inspire?  We don't need to use our imaginations.  We just need to go forth and give it a try.
 

A Tale of Three Classrooms

      If there is a better state curriculum document than the Common Core Georgia Performance Standards, I challenge you to send it to me right now.  I was made aware of it reading a post from one of its developers, Graham Fletcher.  Starting in kindergarten, it includes 3-act tasks, estimation180 challenges, problems with "open middles", and all the goodies an MTBoS groupie could desire.  As my focus turns more towards the primary grades, it has become a go-to source for ideas and activities, and I have barely scratched its surface.
  Looking through the grade 1 frameworks, I was stopped in my tracks by a game called Fill the Stairs.

The premise is simple.  Students take turns rolling  two 10-sided dice and creating a two digit number.  For each turn, the player writes the number created on any stair.  The object of the game is to keep the numbers in order between 10 and 100.  If there is no space to write the number, that player loses their turn.  First to fill in all the stairs wins.

    I felt that the game was better suited, at least at this point in the year, for our second graders.  I made a slight modification to the game board, adding two small spaces at the bottom for place value purposes.  In the first classroom I visited, I explained the rules and started playing a demonstration game against the classroom teacher, with the students acting as "advisers".  After we were confident the kids understood the premise, we set them off to play, first asking them to turn in a sheet recorded in pencil, then asking them to play on blank ones in their SmartPals. 

It was interesting to see their strategies develop as they became more experienced playing the game.  We allowed them to decide which die they wanted to use to represent the tens and which to represent the ones.  So if you rolled a 4 and an 8, you could choose 48 or 84, whichever worked best for you.

   The kids loved it, and as the period came to a close we debriefed with some strategy talk.  I left the game with their teacher to use in a center, or with a guided group of students still shaky with their number sense, but as I reflected on the experience I decided to make some changes as I brought it into second grade class #2.  I replaced the dice with number cards, and further modified the game board.

I removed the rules.  I left a larger space for the cards.  Another modification I wanted to try was to not allow the kids to manipulate the place value of the number.  The first card pulled would always be the tens digit, the second card the ones.  Why?  I was curious to see what would happen.

    I gathered the class in the front of the room and showed them the page on the Smart Board.
    "Today I'm going to teach you a game," I told them.  "But before I do, I'd like you to do something for me."
    Although they had never done it before, I asked them to respond to the following prompts:
   "What do you notice?  What do you wonder?" 
   There was quite a bit of buzzing as they shared their observations with each other.  They noticed stairs, the 10 and the 100, the lines at the bottom of the page, the big space in between the two.  They wondered what the rules of the game might be, if they were going to be asked to count by 10s, why there was a space for tens and ones. 
   Being prohibited from changing the place values of the digits changed the nature of the game.  It took longer to fill the stairs because there were fewer options, but I felt it removed a layer of complexity and reduced the game's potential to build both number sense and understanding of place value.

Many students adopted the strategy of using the second step for the 20s, the third for the 30s, and so on.   What happens now?

   So it was on to classroom #3.  I followed the same procedure, but this time made no mention of the fact that it was a game.  And I asked the kids to write down their "noticings and wonderings" in their notebooks.  Many children wrote that they noticed stairs and numbers, others wondered if they were going to be asked to count by 10s, and whether or not they would be playing a game.
     I need to thank our three amazing second grade teachers for allowing me to turn their classrooms into laboratories.  I am excited to work with them this year, together exploring some different practices and new principles in the attempt to build number sense and make math more meaningful for our second graders.

My Factor Captor Obsession

    Last June I published post detailing my nearly year-long exploration of the game Factor Captor.  Up until the last week of school I had the fourth graders cutting apart the game board and playing around with the numbers:

I was curious: How many multiplication sentences could be made using the 48 numbers on the board as both factors and products.  With three per sentence, could someone make 16?

Close!!  But can you spot the mistake?

     I didn't get any further than this; it was more of a test drive to see what would happen, and if the activity would be worthwhile for next year's class. But as fortune would have it, our grade 5 curriculum opens with a three-week unit on factors, primes, composites, squares, and divisibility rules.  So I knew I would have an opportunity to use the game to extend some learning.
    First, Rich and I let them get reacquainted with the game.  We pulled out the boards they were familiar with.  I was pleased to see that the majority remembered how to play, and for the rest a quick review of the rules was sufficient to get them up and running.  I thought it might make things more meaningful if we explored the unit's vocabulary (prime, composite, even, odd, square), using the numbers from the board.  It seemed like the perfect opportunity to get out the scissors and glue again!
     I thought they might find it helpful to use Venn diagrams.  Many of them chose prime and composite as their first sort.  Of course there was much debate about where to put the number 1.

Some students used reference books.

     Many students finished one sort, and we encouraged them to choose two new labels and try another. And since we're into noticing and wondering, we asked them to write one thing they noticed about their diagram.  These ran along the lines of statements like:

• Most of the even numbers between 1 and 37 are not square, except 4 and 16.
• There are no numbers that are both prime and composite.
• 2 is the only even prime number.

I liked that they were attempting to out into words what they were seeing in their Venns.

And since the obsession shows no signs of abating...

We introduced the advanced grid with a noticing and wondering  "do now".

The kids had some interesting observations, including wondering if playing on the new grid made the game easier or more difficult, noticing that, except for 1, the single-digit numbers are repeated and wondering why that might be so, and wondering whether or not they would get a chance to play.  Well of course!

   I knew that this question would come up, and I knew that, as comfortable as most of them were with the original Factor Captor game board, this one was going to be somewhat intimidating.  I mean, 51?  How would one go about finding its factors?  My hope was to build some intellectual need for divisibility.
   My experience exploring the learning opportunities embedded in the game Factor Captor has me excited about the possibilities that lay hidden within other games, at other grade levels.  It has me thinking more about how we can put games to better use in math class.  In my experience, games such as Factor Captor are used as reinforcements for concepts and skills that have been previously taught. Teachers might provide their students with the opportunity to play them at centers, or when they are done with classwork.  But children who struggle often have limited opportunities to play; it may take them most of the class period to complete their assigned work.  For others, playing the same game in the same way over and over again can quickly become just as dull as another workbook page.  But what if we used the games, not as afterthoughts, or as ways to keep some kids busy while we work with others, but as the vehicles to deliver instruction? Turn them upside down and inside out, take them apart and put them back together?  Not every game lends itself to this kind of treatment, but there are many that will.  I have some in mind; feel free to comment with your thoughts and suggestions.
 
 

Off and Running

  So it's time to get back to work.  I'll be collaborating again with Rich in fifth grade, and we decided to start the year with a problem solving project.  We felt it would set the tone for one of our focal points, which is to help the kids develop their questioning skills.  We would follow the same protocol as last year's movie theater project.

We used this page...

...from this book.

  We gave the kids some time to look at the Birthday Party Basics list, and asked them what they noticed and what they wondered.  We gave them some time to write down their questions.
    While they were working, I overheard two students trying to formulate a question regarding the pints and quarts of ice-cream.  I walked over to listen in, and one student turned to me and said,
   "Mr. Schwartz, what's bigger, a pint or a quart?  I can never remember!"
    I hesitated a bit, and gave her my i-pad.  "Here.  Find it out for yourself."
   "I know how to do that," she said, as she quickly googled "what's bigger a pint or a quart".  (A small moment, I know.  But the fact that she could find the answer herself instead of relying on me or a fellow student empowered us both.  Another reminder of how important those small moments can be.)
 

They chose their favorites to write on chart paper.  Between the AM and PM classes there were quite a few to look through.  Many kids wanted to know how much it would cost to buy everything on the list.  They were also intrigued by the fact that a bag of pretzels cost less than a bag of potato chips but served more people.

A representative sample got posted on Rich's bulletin board.

We decided to have the entire class work on this one.  We liked it because of its openness; we felt it would prompt some interesting discussions as kids prioritized their list, and then re-evaluated decisions  based on how much money was left to spend.

     We weren't disappointed.  The question sparked some good conversation.  There was debate about which size cake to buy, what kind of drinks to serve (one group decided that bottled water was the way to go because kids might be allergic to apple juice and soda, another group agreed but decided to save money by just serving tap water), whether it was necessary to have potato chips and pretzels, whether or not it was important to have money left over, and the difference between 1 pint of chocolate, vanilla, and strawberry ice-cream and 1 pint of chocolate, vanilla, or strawberry ice-cream.
     The AM class worked in groups of three.  Not so great.  One (or sometimes two) tended to dominate while the other(s) were left with nothing to contribute.  You'd think I would have learned this by now.
     We switched it up in the PM.  Rich had them work in groups of two.  As they started in on the problem, I remembered something that I heard this past summer from a teacher at a workshop.  When she has kids collaborating this way, she makes each use a different colored marker.  Her kids know that she is expecting to see a balance of colors.  It helps promote accountability.

I had never tried it before, but it seemed to work!

       Here's what I like about this approach:

• The kids generate the questions.
• The kids get to select which questions they'd like to solve.
• The questions vary in level of complexity and necessitate the use of a variety of skills.
• The kids can work on these at centers, independently, or in guided groups as the unit progresses.

The day the kids were working on their solutions, this tweet came across my feed:

It's like teacher christmas! Except that we buy all the gifts ourselves and they are greeted with subdued acceptance! pic.twitter.com/4Md59dIOW7
— Justin (@JustinAion) September 10, 2014

   I'd like to experiment with big whiteboards for this type of problem solving activity.  Using whiteboards would allow the kids to erase when they make a mistake or want to revise their work instead of crossing out or just tossing the paper and getting another one.  Justin, who knows a lot about whiteboards, tells me that I can get them at Home Depot, and that they'll even cut them to size if I tell them I'm a teacher!
    So school's in.   We are excited about building on all the good work that the fourth grade teachers did with this group last year, and I think we are off to a good start.

My Confession, Part 2: I Am Saved by the MTBoS

    This poster is displayed prominently in my room.

     I tell my students that it's there to provide them with encouragement and hope, but really it's there for me.  It gives me comfort, even though I highly doubt Einstein had the type of dysfunctional relationship that I have had with math.

    Several years ago I was plucked out of out of my elementary classroom, where for over 20 years I taught reading, writing, science, social studies, health, and, yes, math to 7 and 8 year olds.  I put up bulletin boards, made sure everybody got cupcakes and juice on birthdays, planned and took field trips to zoos and museums, drew smiley faces on papers, filled out report cards, did the million things, both large and small, that all elementary school teachers do.  And then one September it was over.
    I was given the title "math specialist", a title that made me cringe, because my mathematical ability is by no means special.   But as long as the math didn't get too difficult, I had no doubts about my ability to help struggling learners.  In fact, I felt I had a small advantage; I could empathize with their struggle and perhaps be able to re-teach and explain concepts in ways that might make sense to them. For inspiration I looked to a familiar world, the world of sports.  I could be like Charlie Lau!

A mediocre hitter himself, Lau (right) is considered the most influential hitting coach in the history of Major League Baseball.  His disciples included Hall-of-Fame third baseman George Brett, a lifetime .305 hitter.

      I spent the first several years in my new position mostly pulling kids, both individually and in small groups, out of their classrooms and back to my room for one-to-one and small group instruction.  We call this "basic skills".  I was patient. I was sympathetic.  I took out manipulatives.  We did journal pages together, maybe got a start on the homework.  We studied for unit assessments.  I did what I could to patch them through a curriculum that raced relentlessly forward and never slowed down long enough for them to catch up, then sent them back into the inferno.
    Two years ago, something happened that changed everything.
   One day,while surfing the internet for math resources, I followed a link to Dan Meyer's TED Talk: Math Class Needs A Make-Over.  There's a powerful sequence when he shows a page from a textbook...

Just looking at it made my eyes glaze over and gave me a familiar sinking feeling in the pit of my stomach.

 ...and then strips everything away until he's left with just the visual of the chairs going up the lift.

   "Which section do you think is the steepest?" he asks.  That was a question I could answer.  It was a question anyone could answer.  You could just eyeball it and make an intuitive guess; you didn't need any "math".  I was hooked: how would we find out?  If there was math that would help answer that question, then that was math I wanted to learn.  Very powerful stuff for a kid who just wanted to crawl under his desk during math class.  I replayed the video over and over, and from there went straight to his blog, started at the beginning, and began reading.  I became convinced that if he had been my math teacher, things would have been much different, and realized that what I had been doing wasn't really teaching.

   Next came Paul Lockhart's  A Mathematician's Lament.

 I remember reading it and thinking, "This must what it would've been like to read a samizdat in the post-Stalin USSR." I imagined math teachers passing worn and dog-eared copies to each other, one step ahead of supervisors waiting to confiscate the manifesto and denounce them as heretics.  I just couldn't believe that a real, honest-to-goodness math teacher would write something that was so damning of his profession and that so accurately captured my learning experience.   I wanted to cry when I read this:

...if I had to design a mechanism for the express purpose of destroying a child's natural curiosity and love of pattern-making, I couldn't possibly do as good a job as is currently being done-I simply wouldn't have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education. 

    It would be hard to overstate what this meant to me.  It meant that I had, somewhere buried deep inside, an ability to do math.  Maybe it was small, but it was something that could be nurtured and, given the right conditions, it could grow.   It had been crushed out of me all those years ago, but with some help it could be found again.
 
   I suppose I followed a well-worn path: from Dan, who taught me about 3-Acts and intellectual need; to Andrew, whose work at estimation180 has had the biggest impact on my practice; to Fawn, whose humor,  humanity, and creativity  has helped me keep my eye on the ball; to Michael,  whose relentless and passionate search for meaning inspire me to dig deep; to Graham, an elementary compadre who keeps me company in a middle- and high school world.  And there are many others.  I took Jo Boaler's course, and learned about Carol Dweck's growth mindset research.  So it was true.  I could learn, not just how to do math, but maybe even to like math.  And if it was true for me, it could be true for all those other strugglers out there: the finger counters and the red x'ers, the fraction flunkies and the long division losers, the times table fist bangers; the confused, the lost, the drowning, and the already drowned.
   So I joined a wild and wonderful community called the MTBoS. I lurked.  I started commenting on other people's blogs.  I started my own.  And I've grown more as a professional in the past two years than in the previous 25 combined.
   There is another, smaller picture hanging in my room.

   This one has been with me since my first years teaching.  But it has taken me all this time to realize that the words apply just as much to me as they do to my students.  We no longer have to use our imaginations to envision what engaging, exciting, and nurturing math classes can look like.  The teachers who are embracing and exploring new ways to make math meaningful in their classrooms are taking no small amount of risk.  But they are doing no less than what they expect of their students.  I am proud of them, and proud to be counted in their number.
    School's in.  It's time to get back to work.