NCTM and the Math Forum

(This post might read like inside baseball for those who aren’t interested in our math education professional associations in the US. If that’s not your jam, feel free to skip this one.)

The news has broken that NCTM is planning to dissolve its partnership with the Math Forum, effectively dissolving the Math Forum, which has been an incredible leader in math education since 1992. As was entirely predictable, the math internet is freaking out at this news, for all the right reasons. I’m afraid this blowback is probably catching NCTM by surprise: it will be a much bigger deal to us than they probably anticipated. I wanted to take a stab at articulating why.

My first annual NCTM was in New Orleans, in 2014. One of my more searing memories was a dinner with a bunch of #MTBoS folks (MathTwitterBlogoSphere–a collection of math educators who share their work and support others online). At that dinner, there was serious angst and even some rage about NCTM. The general sentiment was, Why do I need NCTM when I get everything I need from you all?

Being a member of NCTM is expensive. There are significant annual dues: $124 if you want the journal. And then attending the conference costs thousands of dollars. This coming year, in DC, the conference block rooms start at $289 and go well into the mid $300s, without fees. Registration is $445. And then there’s airfare, meals, etc. Teachers are usually not funded to go to conferences, and some I know have to pay for their sub coverage to miss school (this is insane) and pay out-of-pocket for the conference.

At dinner, teachers wondered aloud about what they get for all that money. An annual conference and a journal was consensus. They didn’t feel much support the rest of the year, via affiliates. They didn’t see evidence of advocacy on the national stage (not saying it wasn’t happening; just that people didn’t see it). And, most of all, they saw zero interest or involvement from NCTM in the thriving online math community, the MTBoS, which was the best source of PD they knew.

There were interesting discussions. I learned the average age of an NCTM member was 55. The average age of people around that table was lower, probably early- to mid-40s. They were also passionate and exciting and innovative team players. They made amazing resources for each other and shared them freely. I remember thinking clearly If NCTM is irrelevant to these people, NCTM is going to die. 

I wasn’t the only one with that thought. There was a lot of talk about what would happen. What obligations do current teachers have to support their national professional association? NCTM had been a leader in Math education for decades, and there’s gratitude for that work, and a desire to pay it forward. One option was to get very involved in NCTM and try to help it grow toward something more relevant for teachers, so membership would begin to rise again.

But did NCTM want that involvement? Or would they roll their eyes at the “kids” and their twitter and do everything they way it had always been done? In New Orleans, that felt like an open question to me. I’m sure I missed loads of nuance, but that was my impression. It was tense.

In 2015, two very big things happened that made that tension dissipate and dissappear. The first was the announcement at Twitter Math Camp that NCTM and the Math Forum were merging. The optics of this announcement mattered. It was at Twitter Math Camp–the conference the MTBoS created to meet their needs.

I can’t speak for everyone in that room or following along online, but for me, this announcement was enormous. At that point, I knew four of the Math Forum people–Max, Annie, Suzanne, and Steve–and admired them so much in every possible way. NCTM’s decision to merge with the Math Forum showed that NCTM was moving toward the future, bringing on people with expertise and relationships in the online world, and recognizing and picking up talent. It was huge. It brought them credibility.

The second thing that happened in 2015 was Nashville. NCTM intentionally invited MTBoS participation on a conference committee, and wow did that make a difference. Robert Kaplinsky shared about his experience online. The keynote was an invitation to join the MTBoS. There was a tweet-up afterward, and Diane Briar came. I remember some of the very same people who were upset in New Orleans saying, “I feel none of that tension now.” We were all excited to see these worlds coming together. To see NCTM becoming more inclusive and inviting and forward-leaning. To see no choice necessary. It felt like our professional association had recognized what we needed, and adapted and evolved. The future looked bright.

Since then, there’s been Math Forum and MTBoS representation on every committee. NCTM offered to host game night, supported ShadowCon, and tweets much more actively. It took a year for the legal teams to allow the Math Forum to work at NCTM, but as soon as they could, they made great and visible changes to NCTM. As just one example, The Math Forum made NCTM Central a thriving part of conferences, drawing many of us into the exhibit hall throughout the conference. We’ve made so much progress.

But now, NCTM is ending it’s partnership with the Math Forum and dissolving its resources and contributions. Institutions like Problem of the Week (POW) and Ask Dr. Math will be lost, which impact tens of thousands of students annually. NCTM owns the brand The Math Forum, and it’s making it go away.

Like many others, I am crushed and angry and worried. First, the people at the Math Forum are some of the most gracious, smart, professional people I know. I mean, I love these people. I am so grateful for all their work, from bringing us together and raising new voices at ignites to the positive, productive attitude they’ve brought to every conference and committee. I can’t believe NCTM is disbanding them. My first concern is for Max, Annie, Suzanne, Steve, Richard, and Tracey.

My second concern is for the future of NCTM. This move threatens all the hard work and relationship-building they’ve been doing. Over the past few years, they’ve built a lot of loyalty among the MTBoS. I have been excited for the future, knowing the Math Forum people are there and working behind the scenes. Now, I honestly don’t know what the future holds.

This move is so damaging. Damaging to trust, relationships, credibility.

It’s not too late to repair, though. I hope beyond hope that the NCTM board will reconsider this decision and repair these relationships. This is a moment to determine the future.

I think Graham summed it up perfectly:

 

Field Science in Education

There’s a lot of angst in education about the role of research. What’s good enough? What’s not good enough? How do we know “what works?” How are some studies considered convincingly quantitative by some but held up as poor use of meta-data by others? Likewise, how are some studies considered robust and qualitative by some, but dismissed as anecdotal and subjective by others?

Honestly, I don’t have answers to these questions. I’m not a methodologist. I don’t have a Ph.D. and I haven’t completed a recent, rigorous course in research methods.

What I do have to offer, maybe, is a geologist’s perspective. I’ll try to explain. Bear with me through this bizarro opening.

I recently agreed to an event in Santa Fe,  which triggered a memory of one of my favorite professors from way back in my undergrad days. Kip Hodges was a hotshot geologist and geochronologist at MIT in the 1990s, but told me once that if geology ever interfered with his relationship with his wife or his daughter, he’d find another way to make a living. He thought he’d be just as happy running a restaurant in Santa Fe. That wisdom stuck, especially coming from a young guy who had just gotten tenure at such a competitive institution.

Fast forward to when I was pregnant with Maya in 2006-7. A former classmate and I randomly ended up in the same prenatal yoga class, and she told me Kip had left MIT and moved to the southwest. When I learned I was going to Santa Fe, I gave a quick google in case he was cooking up green chile shrimp tacos, hoping to catch up with him. He’s in Arizona, alas, but up popped this video in which he talked about the fascinating work he now does with astronauts. I smiled as soon as I heard his familiar North Carolina lilt, punctuated by the way he has always slapped one hand into the other when he wants to make a point. I put my feet up on my desk and settled in for some good nostalgia.

Starting at about 3:05, my ears really perked up.

If you talk to physicists, most of the time, they’ll tell you there is a way that you do science. If you talk to chemists, there is a way that you do science, and it’s usually experimentally based and it takes place in a laboratory. But there’s another way of doing science, which is a more observational way of doing science, a more discovery-based way of doing science, and a more exploratory way of doing science, and lots of times that gets short shrift. Lots of times people say, “Well, it doesn’t happen in a laboratory, it’s not experimentally based, it does not use the scientific method, and so therefore it’s not really science somehow.”

Huh. That sounds familiar.

A few minutes later, Kip gave a quick primer about thinking like a geologist. I actually took a year-long class with him called something like “Field Geology” back in college, so this was familiar and happy territory for me. These three points were central to his description:

Field geology on earth–a lot of it is about multi-scale observing. I look at things closely and I look at things far away. I get different perspectives on things to try and understand them.

 

The sampling that you do when you get materials is a very tactical thing. It actually supports the work that you do. It’s not the fundamental thing. I don’t go into the field in the Himalaya or some place like that and pick up stones. I don’t wander and pick up stones and bring them back. I go and I make observations in the field and I collect samples that are gonna tell me something in the lab, but I collect them very very carefully when I go.

 

The other thing about this kind of science is it’s based almost entirely on inductive reasoning. It’s not like making an experiment. It’s like making observations and trying to cull processes out of those basic observations.

Here’s where I’m going to geek out about geology on you. It’s my favorite science, mostly because of the challenge it provides. The earth does this beautiful thing of trapping its own history in the rock record. But then, thanks to dynamic plate tectonics, the earth is constantly writing over, reshuffling, rearranging, and transforming that rock record. Think about the name “metamorphic rocks.” Every time rocks get subjected to enough heat or pressure–both of which happen when a piece of crust gets dragged down in a subduction zone or crumpled up when continents smash into each other–the rock and fossil records are blurred, smudged, confused, moved hundreds or thousands of kilometers away from the related rock record, sometimes completely erased and rebooted. The vast majority of the rock record has been subjected to tremendous forces, repeatedly—forces that are strong enough to rewrite history at a molecular level–and then hidden below the surface of the earth or buried under vegetation, cities, and water.

Hence the geologist’s challenge! She can’t recreate geologic conditions in a “gold-standard, double-blind, randomized, controlled, replicable experiment.” There are not tens of thousands of fruit flies, mice, or test tubes of centrifuged samples. She can’t make mountains in a lab in a building at a university. In other words, the methods and tools of controlled experiment design are usually off the table. Instead, geologists travel the world and observe, make meticulous records, look for patterns in those observations, create and test plausible hypotheses, and try to disprove theories. It’s a gas, and, as Kip described in his story of Darwin’s training in geology aboard the HMS Beagle, it’s a powerful enough intellectual model to yield the theory of evolution.

Let me be clear. I’m not dissing gold-standard randomized studies. I counted on them every time I went in for chemotherapy during breast cancer (although I also counted on the clinical expertise of my oncologist and her interpretation of that lab-bench research). I yearn for controlled studies when making evidence-based decisions, if results are available. I recently spent an hour rapt, listening to this discussion of a matched-pairs study of youth mentoring programs that blew everybody’s mind, including mine, because the robust data disproved all our hypotheses and wishes. I was glad for the well-designed, well-controlled study.

But here’s the thing. Big, robust, randomized studies are expensive and hard to get, even when you have trained lab staff, grant money, and genetically identical mice from Jackson Labs. When it comes to schools and education and kids and teachers and communities, they strike me as damn near impossible. Reality is just too messy and varied and complex to isolate single variables across classrooms and schools and time and control for them. I’m skeptical of any study that claims tight control in education.

I don’t feel angst about this, though. I never have. It wasn’t until the moment watching Kip’s video that I realized why. I suddenly saw how my time in geology prepared me to write Becoming the Math Teacher You Wish You’d Had.

What did Kip say field geologists use? Multiscale observation. Tactical sampling. Inductive reasoning.

  • I looked up close at specific interactions between one student and one teacher. I stepped back and looked at how whole classrooms and schools worked. I stepped back further and looked at larger historic trends, and then re-examined classroom observations in light of those larger social contexts. I sought to understand by studying math education at different scales.
  • Like Kip doesn’t walk around and pick up stones (lol), I didn’t pick random classrooms. I used my trained eye to select my samples carefully, looking for a range of classrooms that would teach us all something. I planned my traverses with care.
  • And then I looked for patterns. I went through my meticulous records of all those observations–audio files, transcripts, student work, video, notes–and culled out larger patterns. What kinds of questions did these different teachers ask? How did they handle student mistakes? What sorts of tasks did they select?

The whole time I was researching the book, I referred to this work as “my fieldwork.” It’s the term that came naturally, that fit best, but it wasn’t until Kip reminded me about the difference between field geology and lab science that it became clear why that was my approach of choice. Why, when dealing with the complex world of teachers’ beliefs, students’ inner lives, different social and cultural contexts, and deeply flawed outcomes-based data, I went right for a field-science approach. Why I was never a tiny bit tempted to gather pre- and post- standardized test scores on the kids I observed. Why I focused on observation, selective sampling, and patterns.

Observational science is science.

Anthropological studies of classrooms are research.

Quantitative data are not necessarily better than qualitative results, and they’re sometimes significantly worse.

If we only believe the results of double-blind, randomized studies, then we’ll only have evidence for things that can be measured, regardless of how valuable the data. This approach often works fine in labs, but not in geology, and not in social science. As sociologist William Bruce Cameron said in 1963:

It would be nice if all of the data which sociologists require could be enumerated because then we could run them through IBM machines and draw charts as the economists do. However, not everything that can be counted counts, and not everything that counts can be counted.

Sociology is valuable without those charts. Field geology is valuable without those charts. And educational fieldwork is valuable without those charts.

At least to me. But then again, I’m comfortable with this idea of field science. I’ve adopted the habits of mind of a field scientist.

It’s how I see the world.

Daphne’s DREAM: Drop Everything and Math

School’s starting soon, so our kids have started anticipating, wondering, and talking about what might come in fifth and third grade. The other night at dinner, soon-to-be-third-grade Daphne burst into tears with worry about high-stakes testing, timed tests for multiplication facts, and math textbooks. My paraphrase of what she said, approved by Daphne:

That’s not even math to me. I mean, getting answers fast without thinking isn’t even math. I like it when there’s a problem I have to work on and work on and work on, where I get parts wrong and I have to go back to it and figure out what I did, where it might take me hours, weeks, or even months to figure it out. That’s the kind of math I like!

 

The kids are always like, “Why did that test take you so long? I thought you were good in math?” Sometimes I’m the last one done and they all talk about it. But as soon as I know the test is timed, I can’t think very well, so it takes me longer. And I also take a long time because I like thinking about the problems, not just rushing to get the answers.

 

And I wonder, you know how they have DEAR–Drop Everything and Read? It’s a time where you don’t have to do worksheets or sticky notes or read something assigned. You can just read what you enjoy. Why don’t they have it for math? I wish we could Drop Everything and Math. Instead of worksheets or tests or problems, the kids could just look around and see what interests them. We could find a question of our own and ask it and work on figuring it out until we’re satisfied. That’s the kind of math I like, like what I did in Ghirardelli Square.

While she talked, I had a flood of conflicting feelings and questions. Of course, I was overjoyed to hear that Daphne still knows what mathematics is, regardless of her experience in school math thus far. She’s still intact, for now.

I was simultaneously crushed, listening to her expectations for math this year. Oof.

But then there was her love of working on a hard problem over time, of persevering and enjoying the grappling. Sam and I told her it’s a much more important life skill than giving quick answers to fact-recall questions. I mean, this is a kid who likes to face down a worthy challenge. Check her out after a hard part of this week’s ropes course in the Adirondacks.

Triumphant Daphne, problem-poser and -solver extraordinaire.

I was also struck by this idea of Drop Everything and Math. Because, while it’s beautiful, we all know that if a teacher at any grade in a U.S. school told students it was time to Drop Everything and Math, that teacher would face a room full of blank stares. A classful of students waiting to be told what to do. Maybe even a group of students angry about the unclear expectations and absence of directions.

Based on their experience, students have concluded that teachers and textbooks ask the questions and students answer them. We tell students what to do at every juncture, right down to whether we want them to box or circle their answer, or fill in a bubble (completely! With no stray marks!).

If our students would be paralyzed by the suggestion to find a mathematical question of their own to explore, then that’s a call to action. We need to do something different. Students need to learn how to pose their own mathematical problems and questions, not just answer somebody else’s. And I’m talking about genuine mathematical questions, not just word-problem writing.

If you want some suggestions to get you started, I’ve gathered a whole bunch of resources around question-asking and problem-posing on the CH 7 page. A few hits, briefly:

  • 101questions is a super resource. It’s a huge bank of curiosity-provoking images. Ask your kids “What questions come to mind?” They don’t need to answer the questions–just practice asking. If you can only spare five minutes now and then, you can still introduce question-asking via 101qs. (For a recent blogpost about one way to use 101questions and a bulletin board, take a peek at Mrs. Beauchemin’sAre Our Students Really Thinking?“)
  • Notice and Wonder. As Annie Fetter, Max Ray-Riek, and the rest of the gang at the Math Forum at NCTM have been teaching us, removing the question and asking students What do you notice? What do you wonder? opens tons of possibilities.
  • Take a mathematical walk and see what math your students can spot around your school.
  • Suggest students bring something mathy from home and have a gallery walk to discuss where they see math.
  • If your school is equipped with the technology, you can host your own MathPhoto challenge. Ask students to look for lines, symmetry, curves, intervals, etc. Check the linked archives for more ideas.
  • Invest in mathy playthings that spark conversation and delight. Christopher Danielson has you covered on materials, and Kassia Wedekind‘s ShadowCon talk will inspire you to make your math class more playful this year.

Daphne liked the acronym DREAM for Drop Everything and Math. Is hers a pipe dream? Or one that might see the light of day?

That’s up to you, I think.

Starting Out: First Steps Toward Becoming the Math Teacher You Wish You’d Had

I was honored that Middleweb asked me to write a blogpost specifically for new math teachers. Reposting here:

When I was in graduate school preparing to become an elementary-school teacher, my math methods professor, Elham Kazemi, told me it takes five years to become a skillful math teacher. I remember thinking, “Oh no! What about all the kids I’ll have between now and then? Am I going to ruin them?”

Well, the good news is they survived. I think I even did them some good. Sure, teaching mathematics is incredibly complex and I’m a lot better at it now than I was then. I plan to spend my whole career working to become a better math teacher, and I know I’ll never get bored because there is so much new learning to do. Even so, newer teachers have a ton to offer students. I feel proud of and excited by everyone who chooses to become a teacher, and your upcoming students are lucky to have you.

It’s reasonable to set some goals for your development as a math teacher. Be patient and forgiving with yourself while working to get better. Over the years, you’ll build relationships with your students, you’ll figure out how to build a strong classroom community, you’ll grow your content knowledge, you’ll learn how to facilitate conversations about mathematics, you’ll get more discerning about choosing tasks and curriculum for good pedagogical reasons, you’ll become more efficient and focused about gathering and using formative assessment, you’ll anticipate what students might say and do with more accuracy, and you’ll find your teaching style. It will come. But where to start?

In my coaching and my work with preservice teachers, I’ve learned that my square one is always the same: I want teachers to become addicted to listening to students’ mathematical ideas. It might sound like simple advice, but it’s not. Everything else follows. Once we become fascinated by our students’ creativity and ingenuity, we become more motivated to teach math. We enjoy it more, and so do our students. Soon enough, we dive more deeply into the mathematical content so we can understand why our students’ invented methods work. Before long, we recognize patterns in the way students’ ideas progress, and we crave professional learning about the development of mathematical ideas. We start reading, signing up for workshops, going to conferences, joining Twitter, blogging, seeking out colleagues who are as excited as we are to hear the amazing thing a student said or asked in mathematics today. Our curiosity drives us to read the research and find a professional learning community. We aspire to understand, to talk less and listen more, to ask better questions, to make more thoughtful instructional decisions, to support our young mathematicians. We reflect, and learn, and grow.

On the first day of math methods, Elham told us that she was the lucky one who would introduce us to the fascinating world of young children and mathematics. She taught our cohort to listen to children’s mathematical thinking, and be amazed. Pretty much every positive development I’ve made in my math teaching since has followed from close listening. When I feel unsure of what to do, I think, “Don’t just do something; stand there. Listen.”

The rest will come, in due time.

Have a wonderful school year, and let me extend my most heartfelt welcome to this noble profession.

 

Which Comes First in the Fall? Norms or Tasks?

(Reblogged from the Stenhouse summer blogstitute. If you haven’t checked it out yet, take a peek–lots of authors wrote posts!)

I periodically hear discussion about whether it’s better to start the new school year by establishing norms for math class, or to dive right into a rich mathematical task. I’m opinionated, and I’m not shy about my opinions, but in this case, I’m not joining one team or another. They’re both right.

The first few weeks of math class are crucial. You have a chance to unearth and influence students’ entrenched beliefs—beliefs about mathematics, learning, and themselves. You get to set the tone for the year, and show what you’ll value. Speed? Curiosity? Mastery? Risk-taking? Sense-making? Growth? Ranking? Collaboration? You get to teach students how mathematics will feel, look, and sound this year. How will we talk with one another? Listen to our peers? Revise our thinking? React when we don’t know?

In Becoming the Math Teacher You Wish You’d Had, I wrote about a mini-unit Deborah Nichols and I created together. We called it, “What Do Mathematicians Do?”, and we launched her primary class with it in the fall. We read select picture-book biographies of mathematicians, watched videos of mathematicians at work, and talked about what mathematics is, as an academic discipline. We kept an evolving anchor chart, and you can see how students’ later answers (red) showed considerably more nuance and understanding than students’ early answers (dark green).

 

Throughout, we focused on the verbs that came up. What are the actions that mathematicians take? How do they think? What do they actually do?

In the book, I argued that this mini-unit is a great way to start the year if and only if students’ experiences doing mathematics involve the same verbs. It makes no sense to develop a rich definition of mathematics if students aren’t going to experience that richness for themselves. If professional mathematicians notice, imagine, ask, connect, argue, prove, and play, then our young mathematicians should also notice, imagine, ask, connect, argue, prove, and play—all year long.

In June, I saw this fantastic tweet in my timeline.

It caught my eye because Sarah’s anchor charts reminded me of Debbie’s anchor chart, but Sarah had pulled these actions out of a task, rather than a study of the discipline. I love this approach and am eager to try it in concert with the mini-unit. The order doesn’t matter to me.

We could (1) start with a study of the discipline, (2) gather verbs, (3) dig into a great task, and (4) examine our list of mathematicians’ verbs to see what we did. Or, we could (1) start the year with a super task, (2) record what we did, (3) study the discipline of mathematics, and (4) compare the two, adding new verbs to our list as needed. In either case, I’d be eager for the discussion to follow, the discussion in which we could ask students, “When we did our first math investigation, how were we being mathematicians?”

Whether we choose to start the year by jumping into a rich task on the first day, or by engaging in a reflective study about what it means to do mathematics, or by undertaking group challenges and conversations to develop norms for discourse and debate, we must be thoughtful about our students’ annual re-introduction to the discipline of mathematics.

How do you want this year to be? How can you invite your students into a safe, challenging, authentic mathematical year? How will you start?

Representations and Manipulatives and Tools and Things

One of the things I think about is the relationship between teaching math and the physical stuff that goes along with teaching math. This relationship gets distorted sometimes.

For a while, the elementary world got all kinds of swept up in manipulatives. All lessons became “hands-on” because somehow “hands-on” led to “minds-on.” Deborah Ball’s classic Magical Hopes article does the best job I know exposing the flaws in this stance. If you haven’t read it, by all means, click that link and read it right this second.

People sometimes get hung up on tech in the same way. Recently, I had the chance to share some outstanding work Kristin Gray got from her students when she asked them to take out their notebooks and write down what they were wondering about doubling and halving. They’d been working on 14 x 25 = 7 x 50. Check out these conjectures:

I mean, so great.

Both times I shared this work, people oohed and aahed, and then asked the same question: “Could you use tech to do this? Maybe a google doc?”

I have to confess, I don’t understand this question. With paper and a pencil, students were able to shift back and forth between words and numbers effortlessly, much faster than 11-year-old kids can type. If they’d wanted to make a quick sketch or doodle (perhaps an area model, in this situation), they could. They didn’t have to lose their train of thought while hunting through their device’s symbols for ÷ (an obelus, for fellow #wordnerds). The only apps I know that allow students to think and write so freely are apps that turn tablets into $800 notebooks by letting you write on a screen with a stylus.

I kept wondering, what’s the value added there? What’s the rationale for adding tech? What can it do for you that cheapo paper notebooks can’t?

That’s the question I ask myself about tools, in general, whether they require charging or storage in a plastic tote. What will they do for the mathematical teaching and learning here? Sometimes, the answer is not much. Other times, A LOT.

This year, I did something new in my school. My principal and I made it a priority for me to work with our paraprofessionals. These colleagues are overworked, underpaid, undertrained, and almost never supported to go to PD. Yet they’re responsible for educating about 20% of our students–the neediest 20%. So, this past year, my principal worked out a schedule where, one week per month, I had all the paras in two different shifts on Tuesday and Thursday afternoons. Over the year, we did a book study of the second edition of Children’s Mathematics. I chose it because it’s the foundation of everything in elementary math. The book and its embedded videos teach educators how students think about the operations, and how that thinking progresses. They also teach educators to listen closely–and quietly–to students while they think through problems. Given that the paras had been jumping in too quickly to tell or spoonfeed or direct kids, I wanted them to start learning how to tolerate wait time, how to trust that their students have mathematical ideas, and how to listen as a core part of teaching mathematics.

[Update, June 30 2017. I received a note from one of my colleagues–a proud paraprofessional–today. It was a hard note to read, but I am so grateful that she wrote. She taught me a ton. I am leaving the blog intact so you can see what I wrote, and what I learned. Let’s take a look at that paragraph again, but I’ll fix it, and then make changes throughout:]

This year, I did something new in my school. My principal and I made it a priority for me to work with our special education team, including certified teachers, therapists, and paraprofessionals. These colleagues are overworked and underpaid, and there are many demands on their time because they frequently have trainings about specific disabilities and student needs, as well as IDEA compliance, not to mention the meetings required for IEPs. undertrained, and almost never supported to go to PD. Yet they’re These colleagues are responsible for educating about 20% of our students–the neediest 20%–but both the literacy coach and I get less time with them than we do with the classroom teachers. So, this past year, my principal worked out a schedule where, one week per month, I had all the paras the full special education team in two different shifts on Tuesday and Thursday afternoons. Over the year, we did a book study of the second edition of Children’s Mathematics. I chose it because it’s the foundation of everything in elementary math. The book and its embedded videos teach educators how students think about the operations, and how that thinking progresses. They also teach educators to listen closely–and quietly–to students while they think through problems. Given that I’d observed that some of the paras had been jumping in too quickly to tell or spoonfeed or direct kids, I wanted them to start learning how to tolerate work on their wait time (which we all always need to work on). I’d spent a few years working with the classroom teachers, reading the same book (Children’s Mathematics), encouraging them how to trust that their students have mathematical ideas, and  how helping them learn to listen as a core part of teaching mathematics. I wanted the special education teachers, therapists, and mainstream coaches to have similar opportunities to work on their listening.

I also knew that digging into students’ mathematical ideas would allow me to get the parasspecial education team digging into the mathematics itself. One para loves math and I knew she’d be game, but many were reluctant and some were downright hostile to the idea of this year-long focus on mathematics. I had my work cut out for me last fall.

Fast forward to June, and I gotta tell you, I loved my time with the paras special education team. I think it’s some of the most important work I’ve done in the building and I can’t believe it took me so long to get there. We built a safe space and strong relationships and, most of the time, they got more and more willing to try new ideas, wonder about why things worked, and make sense for themselves. I hope they understand that I’m sharing this story partly to encourage my fellow math coaches to think about how they can support their special education colleagues, partly because it helps me make my point about tools, and partly because I’m so proud of the growth I saw over the year, and want to share it. I hope I’ve done the story justice, and I hope, if I haven’t, they’ll tell me. Nothing matters more than the trust and safety we’ve built together.

One observation I’d made over the year was that several parascolleagues openly despised both the area model and the related partial products strategy for multiplication. They didn’t understand why anyone would do that, and were resistant to multiplying any other way than the standard algorithm. As our year drew to a close, I wanted to devote time to multiplication of double-digit numbers and see if I could get anywhere with this animosity toward these two essential multiplication strategies. I knew that if I just drew or recorded the strategies with equations on chart paper, I would lose them. I’d learned that lesson the hard way, and wanted to avoid the shut down I’d caused before. So I needed something. I needed a tool that would unsettle our typical patterns.

In this case, I reached for graph paper. I handed them each a rectangle of graph paper and asked them, “How many squares are there?” Note, I did not use the word “multiply.”

Nobody shut down. Everybody got to work, and I got a great range of strategies, including the ones above. (The graph paper was 14 x 21, if that helps.)

Man I love a variety of strategies. It’s just the best. Now I had a whole range of decisions to make about where to go from here. (If you want a thoughtful discussion about that decision-making process, you need to read Intentional Talk. It’s had a huge impact on me.) There were a number of things I could do, and a number of competing goals in my head. A few of them:

  • I needed to explicitly connect what they did with the graph paper to multiplication.
  • I needed to get them more comfortable with representations of what they did with the graph paper, both in pictures and numbers. Optimally, they’d be the ones recording, not me.
  • I needed to expose some rich mathematics by digging down into one of these, or by drawing connections among a few of them. Which ones?
  • I needed to take this opportunity to highlight the fresh mathematical thinking from some parascolleagues who have had negative histories in math, who started out the year reluctant but dove into this problem bravely, and who still needed support to see themselves “as math people.”
  • I needed them to explain their strategies to one another so they could put words to their own thinking, and listen to and try to follow their peer’s thinking. (This was an ongoing goal all year and we’d made a lot of progress.)

So here’s what we did. Each person explained their strategy. While they did, I asked for volunteers to come up and record on paper what their colleague had done on our chart paper.

I stayed quiet while the person with the marker recorded, and then naturally turned to their colleagues and said, “Is this what you did?” or “What was your equation again?” We were getting somewhere.

In each group, someone had surprised me with their strategy. In the first group, J said she looked at her rectangle and thought, “Way too many to count.” So she folded her rectangle in half:

She thought, “Still too big,” and folded it in half again:

She looked at that and thought, “I can count that and then multiply by four.” The thing is, one of the factors was odd, so quartering led to a fractional result. She didn’t bat an eyelash:

She had a 7 x 10 rectangle, which yielded 70 squares. She then combined 6 half-squares to make 3, and had 1/2 left over. Each quarter had 73 1/2 squares. To figure out the total number of squares, she pinched together the 4 half-squares into 2 whole squares, multiplied 73 x 4 to get 292, and added them together to find 294 squares. She told us that the folding grew out of her comfort with sewing, and she was completely in command of her strategy.

I did the recording on J’s work. Not beautiful, but everyone agreed that what I drew matched what she did.

In the next section, L also began by folding her rectangle in half, but the other way:

Then she groused at me, “Oh, you made it not work out evenly!” A moment later, she said, “That’s OK, I love thirds.” Look how pretty:

 

A perfect square!

When she unfolded it, she had this:

Now we see one power of a tool. If I had explicitly asked them to solve the multiplication problem 21 x 14, I would have had almost all identical column multiplication solutions, which aren’t ripe for rich discussions. But this little graph paper rectangle yielded a wide range of approaches, including two strategies that made beautiful sense, visually, but had almost no chance of emerging if we’d only worked with numbers:

21 x 14 = 4 x (10.5 x 7) and

21 x 14 = 6 x (7 x 7)

Not only that, both J and L were in the spotlight for innovative math, when both J and L have historically not been so keen on the subject. They were the experts on these strategies, teaching me and the rest of their curious peers. If I could have bottled that moment and given it to them for safe keeping, I would have. It was a highlight of my year.

That brought us to the end of Tuesday. I knew I’d start with these on Thursday. Because J and L were in different sections, they hadn’t seen each other’s solutions, although word spread quickly throughout the staff and I heard them comparing notes after school (yeah baby). We began Thursday by marveling over the two strategies, comparing and contrasting the difference it made to want to avoid fractions or not be bothered by them. I then focused us on L’s strategy, written numerically. When we looked at the piece of paper, we could all see that there were six 7 x 7 squares. But when I wrote the equation:

21 x 14 = 6 x (7 x 7)

there was a lot of wondering about it. They all agreed that, if they’d been working with numbers only, they never ever would have transformed 21 x 14 into 6 x (7 x 7). They wondered, where did those numbers come from? Especially that 6? How could a 6 come out of 21 x 14? Eventually, there was recognition that 7 was a common factor of both 21 and 14. That insight led us to write:

21 x 14 = (3 x 7) x (2 x 7).

The 6 was in there somewhere, starting to become more visible, but this is a place where nobody was sure about the rules they’d learned once. What could you do with that 3 and 2? Add or multiply? Five or 6? We went back to the paper:

Do you see a 2 x 3 array there? A 2 x 3 array of 7 x 7 squares? Six 7 x 7 squares? Holy smokes, there it is. They saw where the 6 came from.

We played a little with the quartering strategy in the same way:

21 x 14 = (10.5 x 2) x ( 7 x 2) = (10.5 x 7) x 4

We wrote them up, talked some more about the associative property and what happens when you break factors up by multiplication. Now, to be clear, I am not saying everyone in the group would be able to recreate this logical flow independently yet. But I am saying everyone in the group was following along. They didn’t shut down at the formal math vocabulary, at the symbolic representation, at the diagram.

And that’s why I was glad to have a tool. In this case, the tool made it possible for everyone to access the mathematics here. It helped me gather a variety of solutions so we could make connections among them. It made tangible what had been abstract. It allowed my colleagues to bridge from something intuitive to something a little out of reach. And it made us talk with each other about the mathematics more, not less.

I’d call that added value.

I don’t get fussed over whether tools are high tech or low tech. I love and use them all. But I do take care to use them thoughtfully, not for the sake of using tools or tech, but for the sake of the mathematical learning and conversation they’ll allow me to engineer.

In early drafts of my book, there was a chapter called Mathematicians Use Tools. I was planning to get into all of this stuff. I cut it for the sake of length–it was already a huge book–and I thought tools had been written about a lot elsewhere. They have. I decided, instead, to showcase thoughtful use of tools throughout the book, which wasn’t hard because effective lessons often involved the strategic use of tools. Probably the right call.

There are times I regret not taking the deep dive into tools, though. I see so much tech-for-the-sake-of-tech, tools-for-the-sake-of-tools. I also see teachers still afraid to use tools for fear of mess or noise or lack of control or time or organization. I’d love to explore when and how and why we reach for them–or don’t.

Maybe this is the space.

#LessonClose

I’ve been having fun playing around with lesson closes lately. It’s something I’ve been meaning to write up or present on for a few years now, and one of Mike Flynn‘s awesome Mt. Holyoke classes gave me the reason to do it. I’ll be giving a condensed version of that webinar tonight at the Global Math Department. Register and join us if you’re able. If not, I’ll come back to this space and post the video of the webinar once it’s up, in the next day or two.

Update! Here it is: https://www.bigmarker.com/GlobalMathDept/LessonClose

Some resources:

Slides from the webinar

The list of purposes and formats the Mt. Holyoke crew brainstormed

The planning templates for Open Strategy Shares and the six different Targeted Discussions from Intentional Talk

#BecomingMath 318-321 is an excerpt from my book that shows a close with a different pedagogical purpose

Kristin Gray‘s blogpost, “Number Talks Inspire Wonder

Christopher Danielson‘s blogpost, “What Did You Learn?

Max Ray-Riek‘s googledoc gathering the goodies from MTBoS folks playing with #lessonclose

Someday, I would like to explore these ideas further, in writing or a full-day workshop or something. There’s a lot here! But for now, I hope this helps you get started. I know I’m having fun chewing on #lessonclose.

 

Every year, handouts edition

Every year, I plan to get my slides and handouts done well ahead of time so they can be uploaded.

And then, every year, I think well, at least I’ll find time to upload them during conference week.

And then, every year, I think I’ll upload them as soon as I get home.

And then, every year, I think I’ll upload them before Dan scrubs the conference websites for the files.

And then, every year, I say screw it and just post them here.

Slides: Gut Instincts NCTM

Handout: Gut Instincts handout San Antonio NCSM NCTM

Questions that press intuition: Printable Intuition Questions

 

Finally, every year, I promise myself to get ’em done on time. Next year.

 

NCSM/NCTM My Favorite

I’m starting to bounce back from conference week. I love it every year and this one was no exception. I’m still processing (and digging out from emails), but there’s one thought I need to share now.

Every year there’s a certain amount of soul searching among the folks in the Math Twitter Blog-o-Sphere (#MTBoS) about our role and relationship to our professional guilds. People talk about conferences and communication and membership and journals and the future. All understandable, and a good conversation to have. I agree with some, disagree with others, enjoy listening in. No sweat. Something Michael Pershan said on Dylan’s blog has stayed with me, however, and made me really uncomfortable:

I’ve started wondering if being in the MTBoS is sort of like being a fan. To be in the MTBoS means that you love Three Acts, Which One Doesn’t Belong, Talking Math With Your Kids, Estimation 180, Problem-Based Stuff, Max Ray-Riek, Tracy Zager, Desmos or something.

To be fair to Michael, the next sentence was:

[Is this a good time to remind you that I’m making stuff up and have no reason to believe ANY of this with much confidence? I think it is.]

Ha! I appreciate that Michael tossed his idea out there so we can think about it. Because if he’s thinking it, others probably are too. And I get where it comes from, I think. Mostly. Maybe? I mean, there’s this “cool kids” thing that happens that I can’t stand. So that’s part. And then there are also the amazing resources made by people in the community, (WODB, visual patterns, #3Actmath, Estimation 180, etc.), which many people use and promote. True.

But for me, the MTBoS is not at all about fandom. It’s a community where people who love teaching math and thinking about teaching math can congregate. It’s a place to find like-minded folks, but also dissent. It’s a place where we can make our ideas better by listening to others, putting our work out there, and asking for feedback. It’s a supportive space where some genuine, deep friendships and collegial relationships are born and maintained. It has norms that matter–openness, inclusiveness, camaraderie–even though nobody is officially moderating for those norms. (That’s amazing, if you think about it.) For me, MTBoS is the only place on the internet where the comments are overwhelmingly constructive, and worth reading. It’s not about any one person or group of people. It’s a community, not a cult of personality.

What’s my evidence? There are so many choices (Twitter Math Camp, Global Math Department, every day online), but my recent focus was #MTBoSGameNight. It’s a zany idea, something a few of us hatched before Boston. The idea was to create a way to meet our online community members face-to-face at the close of NCSM and the kickoff of NCTM. We have no budget, no sponsor, no organizing committee. All I do is organize a few logistics and make a slide inviting anyone who wants to come.

Our first year, we had something like 40-50 people? I don’t know. Our second year, we were over 100. This year, 200-250. Matt Larson, as President of NCTM last year, asked how NCTM could help. He asked how could we make sure people could go both to the MET gala and Game Night? Given that Game Night is something the members want to do, what institutional support do we need? This year, NCTM provided the venue and a cash bar. So appreciated.

The night before Game Night, I was talking to Graham Fletcher in a bar and he asked what he could do. I told him I’d love a clever way to give away door prizes. He texted me the next morning. “Where’s the box of books? I’m on it.” He, Zak Champagne, and Mike Flynn set up a series of estimation challenges about them. Now we needed A/V, so Stenhouse helped out there. I arrived at the space 15 minutes before. It was a brightly lit hotel conference ballroom. Muzak was playing. The bartender was drifting around aimlessly. I thought, “This is not going to work.” Nothing about it felt like a party.

And then the people came. Christopher Danielson put his wonderful mathy playthings on every table. People started introducing themselves to each other. Drinks started flowing and the bartender started grinning.

My friend and fellow Mainer, Sarah Caban, came up to me and told me she had an idea. The game she brought was a Rock, Paper, Scissors tournament. Would it be OK if she explained it to everyone? Of course. She got a gaggle of demonstrators, grabbed the mic, and off she went:

Play escalated quickly:

Until the final, glorious climax:

To me, this is the wonder and worth of MTBoS. Somebody has an idea, other people are game to try it, everyone else has a good attitude, and we make stuff (moments, memories, play, thinking, resources) together. Honestly, I felt such an outpouring of love for all these people at this moment. It was the highlight of my conference week.

I don’t see fandom here. Rock, paper, scissors was Sarah’s brainchild, and it was her first game night. She joined twitter about a year ago. The rock, paper, scissors finalists were not Dan Meyer and Andrew Stadel. They were two women I’d never met before. I hope our winner will remember what it felt like to have all those math teachers at her back, chanting, “Dana! Dana! Dana!” Because to me, that’s what we do every day. We have each other’s backs. We cheer each other on. We share our passion for this work, and our ideas, and our energy.

Sign me up to be a fan of that.

I’m pleased to say Matt and I have institutionalized this event going forward. #MTBoSGameNight will continue to be hosted by NCTM. That may sound like a small thing, but it’s not. From where I sit, the merger with The Math Forum, the emphasis on #MTBoS representation on all NCTM committees, the MTBoS keynote in Nashville, and the support of our fringe events like Game Night and ShadowCon are meaningful. I’m all for pressing NCTM to be what we need it to be. But I’m also all for recognizing the big shifts made over the last few years. Change comes faster within the (unstructured, unregulated) #MTBoS than it can within the (highly structured, institutionalized) NCTM, just by the nature of the beasts. But good change is happening nonetheless.

 

Welcoming Dissent

It’s been so gratifying to hear from people enjoying Becoming the Math Teacher You Wish You’d Had all around the world. After five years of work on it, I’m over the moon that teachers are finding it useful, approachable, and inspiring (their words, not mine, but oh boy do I love those three words).

I’m worried, though.

I’m worried about the normal, human tendency to not want to hurt my feelings. I’m worried I’m not hearing from people who disagree, or think, “OK, maybe. But what about…” or “In my experience, actually…” I’m worried I’m missing out on my chance to learn from your critiques.

So I wanted to make it explicit. I’d love to hear how, when, and why you disagree or are unconvinced. You can tell me in the comments, on the forums, on twitter using #BecomingMath, via email, on the (nascent) facebook discussion page, or in person next time you see me.

Of course, please keep it civil and constructive. No need to tag in or poke a stick at the ideologues and name-callers from the math wars. I’m not that in need of dissent.