Thursday, April 23, 2015


So, how do you actually help teachers become aware of and work on their decision-making? I suspect one of the reasons teacher educators/professional developers/coaches/administrators so often resort to checklists and best practices is because they're easy: demo the strategy, explain its benefits, show some examples and non-examples, make a ppt, give teachers 5 minutes to practice it in a decontextualized and artificial setting, and go back three weeks later to evaluate on a rubric. Decision-making is much harder to see and work on because it is so often invisible, even to the teacher who is actually making the decisions.

One of the several pedagogies we've used to support novice teachers in working on their decision-making is called rehearsal; the seminal article from Magdalene Lampert et al is here, and a primer on what and why can be found on UW's Teacher Education by Design site here (click on "introduce"). UW partnered with us last year to redesign our summer training institute (click for video if you're curious about the redesign), which is how I was introduced to rehearsals. I'd encourage you to take a moment and skim those resources, because we're using the term rehearsal to mean something very specific-- it's not just getting up and trying out a piece of a lesson in front of others, or running through or practicing some teaching, but rather, a method of deepening teachers' attention, supporting their thinking, and expanding their repertoire through a laboratory setup.

Leading rehearsals is really hard. I insert this here because they're pretty sexy-- just about every time I've described them to a fellow teacher educator, the response I've gotten has been an enthusiastic "oh, I could totally do that and I'm going to go try it right now!" I don't doubt that they're doable-- after all, I learned under pretty strained circumstances and like to believe I do a decent job-- but I do have a few words of caution about common pitfalls that I have experienced and have also seen from others who are just starting to use this pedagogy:
  • When to pause: I have seen facilitators interrupt teachers almost constantly, which breaks a teacher's concentration, or pause for things that are unrelated to the focus of the rehearsal. There are so many possible opportunities to stop and either unpack what's happening or share thoughts, and it takes quite a bit of discipline to let things go (the classroom analogue: too many teachable moments lead to tangents); without this discipline, teachers end up being inundated with disconnected bits of advice that are difficult to follow and often overwhelming. On the other hand, I used to err on the side of letting too much go-- because I'd spend too much time deliberating whether something was worth pausing for or not, and by the time I decided, the moment was gone.
  • How directive to be: When do you say "do this" vs. offering several options to choose from vs. asking the teacher what they think they should do? When do you ask for a teacher's rationale vs. assume it vs. assume it doesn't matter? There are clearly social consequences to these choices, in terms of how the teachers feel (supported, challenged, threatened, dismissed, validated...), but there are also pedagogical consequences. Simply telling teachers what to do doesn't build their judgment (rather, it starts to create another one of those "good/bad" lists I critiqued previously), but letting novices in particular wander blindly isn't productive either.
  • Involving the group: While the teacher rehearsing benefits from rehearsal, so should the entire group; the other participants aren't there just to play students, but also to refine their own skill at noticing choice points and building a repertoire of possible decisions they could make and what consequences might result from those decisions-- and in doing so, contribute to each other's learning as well. As a facilitator, how often and in which circumstances do you open up a question for the group to consider together? I have missed opportunities to engage and push on participant learning in this way, and I have also opened up conversation in moments that have led to the airing of inappropriate suggestions, unproductive mindsets, or tangents that would have been better surfaced/addressed in a different space.
Here's a video of me leading one of these things-- both to illustrate, but also so y'all can give me some feedback-- as a demonstration at an internal conference earlier this year so that our staff could see the way we used rehearsals last summer (the audience and the "teacher" in the video are former teachers and current administrators/coaches, not novice teachers). The actual rehearsal-- of a secondary number string-- starts around the 4:00 mark; prior to that, the wonderful Emily Shahan (from who I learned all of this) provides some context around why and how we rehearse (including the caveat that this is not a performance).

(password: nola)

Apologies for the mediocre AV quality, but if you were able to make sense of it, here's what you probably saw me do in an attempt to demystify and make explicit the teacher's choices and rationale and to support the learning of both the teacher and audience:
  • Open by naming a set of practices and moves that we would focus on in this particular rehearsal (which, for context, we'd been focusing on all day as we learned about number strings)
  • Preemptively remind teacher of useful moves (to open the lesson by sharing rationale, to use gestures to highlight the area model)
  • Ensure that the first time I pause the teacher, it's with affirmation
  • Name and validate teacher actions (in this case, to offer a model and to give "students" time to discuss the model offered) and provide rationale for why these choices support student learning
  • Reinforce mathematical language (the precision of base and height, solve vs simplify)
  • Ask teacher to share her rationale for particular choices (why was that a good time for a turn-and-talk?), and use that to make a broader point about turn-and-talk
There are a number of facilitator moves you don't see in this clip: pausing to invite the audience to engage in a discussion about a question or a teacher choice, for example, or prepping certain audience members to play particular roles (either in terms of the mathematical thinking they demonstrate or the behavior they demonstrate, depending on the focus of the rehearsal). I know some of y'all out there have also led rehearsals, likely with much more sophistication and intentionality than is visible in this example (and authenticity, given the context here)... can I convince you to share some video for our collective learning too? Those of us in the teacher educator role last summer didn't have time to watch each other, because we were always facilitating at the same time, and yet (just as with teachers), I think there's so much we could learn by observing each other's practice!

Tuesday, April 21, 2015

non-judgmental noticing

I've thought of my work this year as being undergirded by three primary orientation shifts, all of which run counter to our current (predominant-- there are certainly important pockets of dissent) organizational philosophy around teaching and therefore teacher training:
  • From teaching as best practices to teaching as decision-making, which I wrote about here
  • From learning as transmitted to learning as socially constructed, which I've referenced in the posts about the purpose of student talk, here re: the creation of mathematical knowledge, and here re: students' funds of knowledge
  • From +s/Δs to a non-judgmental stance, which I'm about to write about now
Too many conversations about teaching and the work of teaching are about good teachers vs. bad teachers; see: Hollywood, political discourse, ed reform, and the tons of other educators/thinkers/scholars who've written about this unhelpful dichotomy before me. What that creates, however, is a paradigm within which it's incredibly difficult to change. Teachers or administrators or even professional developers who subscribe to this belief (or feel obliged to because of the culture of the school, organization, or society they're in) hold up certain paragons, distill their practice into checklists, and then evaluate teachers based on how many of those boxes they can check. If they had a large sample size of paragons, they can even say their checklist is "research-based." Did you use turn-and-talks at least three times in your class period (regardless of what you had students actually talk about)? Did you offer at least two practice options on your worksheet (regardless of whether students actually choose those options or whether they're both mathematically meaningful)? This turns the work of teaching into something a well-programmed robot can do, and robs the teacher of autonomy, opportunity, and humanity. 

When this is the collective assumption about what makes good teaching, administrators/coaches walk into a classroom looking for plusses and deltas, strengths and weaknesses, glows and grows, etc. No matter what cutesy language is chosen, however, the teacher is on trial and the observation is essentially a sorting exercise with two vague and mutually exclusive categories. But teaching is incredibly complex, and what's "good" in one situation may be terrible in another. A particular choice might have both productive and counterproductive consequences. And, to get better, teachers need both moral support and support around the decisions they make.

So that's a little about why we should shift to a non-judgmental stance, and here's one small way how: I've used this exercise several times over the course of this year with small (~10-15) groups of teachers and teacher coaches, and gotten pretty positive feedback each time. The agenda is simple:
  1. Watch a short video clip of teaching together-- I find that no more than 5 minutes are really necessary, and the brevity also keeps this exercise manageable. Watch the first time to orient to what's happening, and then the second time to write a list of everything you notice (from the simple "the teacher is wearing blue" to the more detailed "three students raised their hands after the first question"). The only criterion for your noticings is that they need to be factual, provable statements (so "the students are bored" doesn't count, but "several students have their heads down" does).
  2. Share all the things that were noticed (in virtual spaces, I set up a Google doc and let everyone type in the same box simultaneously, which saves time and elicits more responses than doing it out loud). The goal here is simply to create a laundry list, but as it's created, note whether you also noticed the same things that others are sharing, or if others noticed things you didn't (in a virtual space, I have people star or add a check mark next to things they also noticed). 
  3. Choose one of the noticings and identify several ways it could be interpreted; the goal is to brainstorm as many interpretations as possible (again, Google doc works well here in virtual spaces). For example, a noticing might be that the teacher wrote both on the whiteboard and on a smartboard. Some interpretations could be that the teacher is trying to organize more important information on the smartboard and more temporary information on the whiteboard, which gets erased routinely; the teacher is simply writing wherever there is more space; the teacher is told s/he has to use the smartboard because the district has invested in it but doesn't really know how to leverage it. This is a place where the facilitator needs to exercise some judgment and choose a noticing that is likely to be rich enough to elicit a breadth of meaningful and plausible interpretations; I find this works best when it's clear the teacher has made a decision, consciously or not (and also sets up the next part of the exercise well).
  4. Ask what the teacher could have done instead, and what the implications of that move would have been. For example, the teacher could have chosen to write things on the smartboard that students should copy into their notes and work-in-progress on the whiteboard, which would have signaled to students what was important enough to record permanently. The teacher could have chosen to provide individual copies of the table for students to fill in instead of creating it on the whiteboard, which would have created visible accountability for students. The teacher could have written everything on the whiteboard but used different colors to highlight different information, which would have directed students' attention to the most important numbers. 
  5. Reflect on this exercise by asking what felt interesting about it, or how this felt different from a "typical" classroom observation, or what participants are now thinking that they weren't thinking after just watching the clip for the first time, or what they became aware of about their own predilections when observing teachers, etc. I also like using abstract sentence starters here, such as "teaching mathematics is..." or "I am thinking... I am feeling... I am wondering..."
Why do I love this exercise as an introduction to non-judgmental noticing? Three reasons stand out:
  • Participants are often surprised by the diversity of noticings/interpretations and things people pay attention to, which reinforces the limitations of relying solely on yourself to improve your own practice, the absurdity of the idea that there is a single "right" teacher action in any given situation, and the importance of collaborating with other professionals. 
  • It also often leads participants to have aha moments about their own biases (e.g. "wow, I was focused on the teacher the whole time and didn't pay attention to what the students were saying" or "hmm, all of my noticings were about management and culture, and others emphasized mathematical thinking much more"), which helps them be more generative (and generous) when they observe teaching in the future.
  • AND if we believe that decision-making is developed through attention to teachable/coachable moments (which I do, and will write about tomorrow), then this exercise primes participants to start paying attention to discrete moments in an episode of teaching that may or may not be worth discussing further.
Want to read more about noticing, and why it's hard/important? Check out anything by Miriam Sherin and/or Elizabeth van Es; their focus (at least, in the articles I've read) is on using video with teachers in professional development, but they've got a lot to say about noticing (and what novice vs. experienced teachers/coaches are likely to attend to) as a result.

Monday, April 20, 2015


Hi. It's been kind of a surreal spring in my personal life, my professional life, and at that odd intersection of the two. Some happenings have been deeply frustrating; others, deeply joyful. I've been spending a lot of time wondering when x or y will settle down, either because I'm tired of uncertainty or because I'm afraid of sustained happiness; I like stability.

And I have been thinking about mistakes recently-- or rather, it'd be more accurate to say that a confluence of moments over a 24-hour span last week (hurray for NCTM bringing so many brilliant people to Boston so I could finally meet them in person!) have connected a number of disjointed ideas that have been floating around in my mind for a while.

I got to hear Harold Asturias speak at the Boston Teacher Residency, because BTR has been so wonderful and generous in letting me follow them around at multiple points this spring (do you know @GraceKelemanik and @AmyLucenta? If not, you should. They're doing some really incredible work in thinking about math teacher education and school partnerships), and he started us off with the video of @dlaufenberg's TEDx talk on learning from students, embracing failure, and letting go of a culture of one right answer (online and IRL worlds colliding!). Here's what we said about mistakes-- thoughts that seemed to be worth capturing all together in one place:
  • It's important to embrace them rather than avoid them
  • A safe classroom environment is required if students are to embrace mistakes
  • Mistakes aren't just tolerable and okay (or permissible and forgivable), but essential
  • And they're essential not just because we have to surface them in order to fix them, but rather, so we can build from the thinking they represent
This last part feels most important-- and most paradigm-shifting-- to me. It's not just that we need students to make mistakes so we can see their misunderstandings in order to correct them, but rather, that student thinking is central to student learning. I wish this statement were as obvious as it sounds. And I wish I could articulate what I mean by that without resorting to jargon. Roughly, I mean that learning happens when students acknowledge, revise, build on, extend, reshape, etc. their thinking, and so knowing what students are initially thinking is critical for any teacher who hopes to influence their thinking. Sometimes, that initial thinking doesn't align with what the teacher might consider "correct"-- not because the student is dumb or because his/her brain is empty, but because the teacher has information or perspective the student doesn't yet have, or is using logical/critical processes to evaluate ideas that the student is not yet using-- and that's why I don't like to talk about misconceptions (@mpershan sent me this related paper last time; I haven't read it yet but plan to soon!). What we call misconceptions or mistakes are so often more like not-yet-understandings, or partial understandings, or even not-quite-deep-enough-understandings (I think a lot of procedural errors fall into this camp, although some do just stem from distraction). 

It reminds me of what I wrote about the purpose of student talk after reading Megan Staples a few months ago (if you read that, make sure to read @doingmath's comment too, which highlights something important that I missed the first time around): are we having kids talk for the sake of talking or because learning can't happen if they don't = are we creating space for student mistakes so we can fix them (whack-a-mole) or because they're an inevitable part of the thinking we do need to create space for?

Later that evening, at the #MTBoS game night tweetup, I got to sit with @delta_dc, @literacygurl, and @davidwees to talk about teacher education. We raved about BTR's model, and also discussed the challenge of working with teachers who are afraid to make mistakes or be judged (although @ekazemi did say, here, that "you can't look good and get better at the same time"). There are so many reasons-- not only is it scary to open yourself up to others, and not only has teaching historically been a closed-door profession, but we're also in an era where teaching practice is super high stakes (VAM/evaluations/tenure), where teachers are blamed for every educational ill, and where we add insult to injury by challenging their commitment (lazy/stupid/don't care about kids/in it for the summers) when they're already overworked, underpaid, and not respected.

The nuance I'll add here is that I work with teachers who are often willing to let us into their classrooms and eager to get feedback-- on what they did "right" and what they did "wrong." Maybe this mindset is more conducive to learning than being completely resistant to observation and coaching, but what still feels tricky to me is that it's still indicative of a belief that there are right answers in teaching; some decisions are "good" and some are "bad," and the way we get better as teachers is to get closer to some arbitrary exemplar. We so rarely have the opportunity to talk about the practice of teaching in non-judgmental ways...

that I think I'm going to have to write a few more posts sharing some of the work I've done this year in trying to shift our organizational culture towards one that is built more on respecting the intellectual work of teaching / the agency of teachers and less on mimicry. Stay tuned.

Wednesday, January 21, 2015

equity, the sequel

Given all the reading and thinking I've been doing, I'm wondering if what I need is a more fundamental definition of equity: what if I were to conceptualize equity as a state in which all people in a group (society, writ large; a classroom, writ small) are valued-- in which they're seen for who they are, heard and understood as stakeholders but even more so as contributors, supported in their goals, given dignity, etc.? If so, then I think our work as teachers is twofold:

1. Create classrooms in which that is true (and serve as a microcosm of this vision of an equitable broader society): in which students (and the teacher) see each other, understand each other, support each other, and learn from each other, both in terms of what they bring into the classroom and in terms of what they create/do once they're there.
  • To do this, teachers must create what Esmonde (2009) writes about as intersubjectivity: a shared meaning of a situation, or a meaning that's as shared as possible. Through classroom culture, through structures, through relationships, teachers develop a space in which students actively seek to understand one another (even if they disagree with one another), and engage in related behaviors such as asking clarifying questions, building on ideas, inviting one another to speak, etc. It's Staples' idea of creating common ground, and central to the model of a participatory democracy in which well-informed citizens are able to create the world they aspire to live in, rather than just being acted upon by those with more power.
  • For intersubjectivity to exist, teachers must pay attention to the intersection of identity and ideas, because they so deeply influence each other. 
    • Our identities (both the more macro identity markers such as race, class, language, gender, ability, etc. and the more transient identities that we negotiate in the moment depending on our context) influence the experiences we have, because of how people respond to us and what they expect from us-- whether they expect us to have good ideas in an academic environment or not (Nasir and Shah write about how African-American boys interpret racialized narratives about who is good at math, particularly in comparison to Asian-Americans)-- and they also directly influence our ideas because these experiences shape how we understand the world and how we think and reason. While several of the pieces I've read about identity and status in math classrooms focus on ideas of smartness and competence, we must also be explicitly conscious of race and other social identity markers. Two primary reasons occur to me: 1) we cannot create equitable environments without being sensitive and responsive to the ways in which our identities-- particularly visible and marginalized ones-- have shaped our life experiences and therefore the beliefs and behaviors we bring into a classroom, and 2) because our socially marked identities are a rich, important, fundamental part of who we are and ignoring them means seeing only part of us, shortchanging what makes us who we are. In the article cited above, Esmonde summarizes this as saying "identity-related processes are just as central to mathematical development as content learning."
    • When we have good ideas, which we gauge through signals from others about whether they value our ideas, we develop identities as people who have good ideas; people who are smart, people who are competent, people who are valued. And it's particularly important for students from traditionally marginalized backgrounds to develop those identities to counteract the messages they're receiving from the broader society on a regular basis.
Esmonde cites a number of interactions that are "more likely than others to lead to meaningful mathematical learning," the types of things you'd aspire to in a student-centered, inquiry-driven classroom: discussion (not just show-and-tell), productive group work, etc. But, students' positioning (which is based in part on their identities) expands or limits their ability to participate in and access these activities, which in turn expands or limits their ability to negotiate their identities as successful learners and thinkers.

2. Use such classrooms and the affirming space they create to explicitly discuss and address why this is so hard to do as a society more broadly, given what has happened in history, given our psychological and cognitive biases, given power structures that are difficult to change, etc.
  • To do that, teachers must be aware of and create connections between what happens in a classroom and what happens outside it; for example, a breach of classroom norms could be an opportunity to discuss what happens when societal expectations are thwarted, using any number of examples, depending on students' interests and on current events: when someone doesn't follow gender norms and is shamed or bullied; when a relationship doesn't fit mainstream images of "normal" and the partners are harassed; when people and institutions we're supposed to trust-- who are supposed to protect and serve us-- instead become the ones who we fear most. Where does the norm come from? Why do people respond the way they do when they sense a breach, and how does that compare to the way they should respond? Does it matter who's breaching the norm? How egregious the breach is? Where or when the breach happens? Consider the ways in which we've addressed the breach in our classroom; could that process work outside of our classroom? 
What do you think? What's my definition of equity missing, and what else can/should teachers do in its service?

Tuesday, January 20, 2015

bridging children's knowledge bases

For a related but slightly different take on (what I'm calling, because this is my mini-research project) culturally competent teaching, I read Turner et al.'s 2012 paper, which uses a framework that I find very similar to Lefstein and Snell's (described in this post): attending to particular aspects of students' thinking or knowledge, developing a particular awareness-- or interpretation-- of what students think or know, and then eliciting that thinking or knowledge. They apply this framework to two types of thinking/knowledge in particular: students' mathematical thinking and reasoning, and students' cultural/home/community-based knowledge and understanding.

My first question is whether the two are necessarily distinct-- my sense is no, and implying that they are necessarily distinct implies that students' cultures/homes/communities are not mathematical, and mathematics doesn't exist in their cultures/homes/communities. I don't think that's what Turner et al. are trying to imply. However, it does raise a question for me of where you draw the line-- is it that the two exist in a blurry Venn diagram, such that some types of thinking or knowledge are just mathematical, some are clearly not mathematical, and there's some overlap?

Regardless, the literature (and common sense, one would hope) suggests that bringing students' cultural/home/community knowledge (also known as "funds of knowledge") into the classroom is a good idea for two reasons in particular: 1) affirming and leveraging students' funds of knowledge makes them feel valued/welcomed/included in a classroom and 2) it helps them make sense of school mathematics that otherwise can seem foreign or intimidating by providing a schematic hook, so to speak.

Turner et al. also write about how school mathematics (or formal mathematical thinking) and cultural, home, and community-based knowledge and reasoning (or "informal" mathematical thinking) can be bridged in superficial ways-- like I wrote about here; they call these emergent connections-- or in more substantive ways-- which they call meaningful connections. Meaningful connections are grounded in understanding students' non-school lives rather than assumptions (emergent connections might be too, but meaningful ones have to be), exist in contexts that are familiar and common, AND use mathematics in an authentic way (remember: no student takes a given dollar amount and tries to determine the total number of combinations of three separate snacks that can be purchased with that exact dollar amount).

This brings me to a hypothesis: can we say that the thing that makes "good teaching" (as implied over the past several posts-- instruction that attends to and values and is designed around student thinking in such a way that makes students feel competent, affirmed, and at the center of the classroom) culturally competent, in addition to just being "good teaching," is this explicit leveraging of students' funds of knowledge? After all, it requires the type of deep relationships and deep understanding of students' individual and collective identities and interests that are a hallmark of culturally competent teaching.

And it brings me to a question: when we're talking about bridging school mathematics with students' funds of knowledge by selecting or creating tasks that are locally contextualized and authentic, we're talking about curriculum: the things they know. That seems fairly straightforward to me. But what would it look like to make a pedagogical bridge: to bring in students' ways of thinking, communicating, and interacting (how they know and learn), too? I can think of some ways in which this makes sense: outside of school, children tend to talk and laugh and ask questions, so they should be able to do that in school too. But I also know that in many homes, particularly for children of color and children whose parents may have had limited formal education or are working class rather than professional class, children are expected to unquestioningly obey authority-- in fact, that may be how they're taught to survive a world where even doing exactly as a police officer asks could still result in your untimely death if your skin is dark enough.

The type of pedagogy we've been describing as desirable is the exact opposite: where students think for themselves, ask questions, and don't unquestioningly internalize what the teacher says, and where the teacher's authority is limited. And I'll stand up for this type of pedagogy, because I think it's ultimately more affirming (even though I recognize as I say that that I may be implying that authoritarian parenting styles are not affirming), and because I think it's what students from traditionally marginalized backgrounds need and deserve for education to be liberatory. But then, aren't we contributing even further to the wedge that some students feel education places between them and their families? How do we talk with them about that?