Thursday, December 18, 2014

what's the purpose of student talk? collaborative inquiry

I've been traveling what is essentially a four-weeks-on, one-week-off schedule since August, which is doing a number on my socio-emotional well-being, my personal life, and more importantly to you, my online presence. While I'm normally a fairly voracious airplane reader, all I've been able to do above 10,000 feet lately is nap and watch trashy tv (thank you JetBlue), because even catching up on blogs/twitter makes me feel overwhelmed. I did, however, manage to read Megan Staples's 2007 article, "Supporting Whole-Class Collaborative Inquiry in a Secondary Mathematics Classroom," on Lani's (@tchmathculture) recommendation, and it was worth every word-- by which I mean, I'm kind of obsessed with it. If you're looking for a summary, check out Raymond's (@MathEdnet) here, because I'm not even going to try. Instead, I'm going to share an assortment of induced musings, with the (dim) hope that someday I'll be able to come back and flesh out these ideas further.

What's the purpose of student talk? - This isn't a new thought, and one that I'm sure has reverberated around the #mtbos and one I've read about in Smith and Stein, among others: why do we ask children to talk in math classrooms? It's easy to identify culture/management reasons-- it keeps kids "engaged" (I could write a whole other post on that word, and @delta_dc has a lot to say about it too), not asleep, less bored, whatever-- to claim efficiency (students can ask their peers questions instead of waiting for the teacher), or to resort to tautological reasons (because discourse is important), and much harder to dig into why it matters for students to talk. But why it matters has implications for how we set them up to do it, so let's talk about why it matters.

If it just matters for the reasons I've already listed-- which I don't think are completely insignificant-- then we can get kids talking by mandating it (cold calling, participation grades), by instituting roles, by instructing them to "work together." And what often happens is that students work in parallel: side-by-side, occasionally asking or answering questions, but just enough to get the work done. Have you heard of LessonSketch? It's a pretty cool tool created by Pat Herbst and Dan Chazan and their teams, and it's what allowed me to throw together this little illustration in less than 10 minutes:


Reading the article, however, is helping me articulate that I believe talk matters because learning is social: we construct stronger (better and also more deeply understood) ideas with others than we could arrive at independently. This happens through several mechanisms (which have undoubtedly been researched and laid out by smarter people than me), including:
  • By having to express their own ideas to others, students clarify and refine their own thinking
  • By listening to others' ideas, students evaluate (and add to, revise, or confirm) their own because they may be confronted with contradictory perspectives, different rationale for the same conclusion, other ways of explaining the same idea, etc. 
  • By connecting their ideas with others' ideas, students build something more inclusive (or generalizable or deeper or broader, depending on the goal of the conversation) than what they developed independently
To move from working-in-parallel to this type of co-construction requires really skillful facilitation on the teacher's part, which is where Staples' article is so brilliant. She posits that there are three primary functions the teacher plays:
  • Supporting students in making contributions: this goes beyond simply eliciting students’ ideas—the ideas need to be valuable contributions to a conversation, and the teacher can help find the worth in each idea. By finding the logic in a thought that initially sounds incorrect, or asking specific follow-up questions (rather than just “say more”) that require students to elaborate on the key concepts they’re describing, the teacher both elicits fodder for discussion and reinforces the message that it’s not just enough for students to explain such that the teacher understands, but rather, they need to explain so their classmates can understand. Everyone needs to have access to the content if they’re going to be a full, equal participant in the building of it, and what allows everyone to gain access are the contributions each student makes.
  • Establishing and maintaining common ground: you know those conversations where it feels like everyone’s just talking past each other? Political disagreements with extended family over Thanksgiving dinner, perhaps, or Q&As where the questioner is more interested in communicating his/her own point than actually asking the speaker a question? Establishing and maintaining a common ground is what the teacher does to ensure that students are actually having the same conversation rather than just pushing their ideas into the same space (think of a series circuit rather than a parallel circuit). Directing students’ attention to something worthwhile, stating what has been figured out so far, and maintaining a record so that all students can follow the conversation allows the class to build collectively from one student’s idea rather than simply taking a survey of what everyone thought and then picking the thought that’s correct.
  • Guiding the mathematics: these collaborative discussions build students’ participatory, social, and dialogic skills, certainly (and very importantly), but we’re still in a math classroom, and students are developing particular mathematical ideas. Although they’re co-constructing their understanding of a topic, they’re not inventing that topic from scratch; they’re figuring out something that has, by and large, been figured out by others before them. So, the teacher has to know the ideas that are most important to a topic (and common patterns and progressions for their development) and then choose appropriate tasks and constantly assess and respond to where students are in their thinking.
Within these three functions, a few underlying principles or stances stood out to me as being most distinct from what I typically see in classrooms:
  • The teacher doesn’t attempt to protect students from hearing “wrong” ideas. This strikes me as beautifully respectful; when we shield students from hearing “wrong” ideas because we think they’ll be confused or because we fear they’ll believe something untrue (good intentions, certainly), we’re essentially saying we don’t trust them to think for themselves, or that their grasp on reality is so tenuous that they won’t be able to realize that something is incorrect, or that if they do latch on to something false, they won’t have the wherewithal to recover. Relatedly, the teacher doesn’t send a simple “mistakes are okay because everyone makes them” message of tolerance, but rather, she embraces mistakes—not just because they allow us to correct ourselves and so we inevitably learn from them, but also because entertaining/weighing/invalidating discrepant information deepens our understanding and strengthens our conviction.
  • There’s an example where one child calls another stupid, and the teacher’s response isn’t to quickly cut him off with “that’s not nice” or “we don’t disrespect each other in this classroom.” Instead, she demonstrates the value of the first student’s thinking by asking him to turn his confusion into a question, and then later validating that the class wouldn’t have reached its conclusion without his question. In this approach, the name-caller isn’t reprimanded or silenced (or made to feel like an outsider who just broke a norm), but rather is shown why calling his classmate stupid was wrong. The student who volunteered the idea is shown not only that he’s not stupid, but also that his thinking is essential to the shared work of learning. 
  • Staples writes about the use of representations as a scaffold, rather than the use of representations for representations’ sake. Duh. But I so often hear (and have said), in response to the (stated or implicit) question of why we are using graphs and tables and diagrams and equations and words, that “it’s important to know all the representations and translate between them because then you know all the ways something can be represented” or even “because some representations highlight mathematics that others don’t,” which are—though not untrue—different from “because doing so gives us access to more of the mathematics.”
  • When the teacher attends to pacing, it’s not because of “urgency” or in order to “get through” the whole lesson, but rather to control the flow of information so that every student can process enough of what’s going on to maintain a foothold in the conversation. This doesn’t need to mean that every student understands everything that has been said at all times—just that every student is comfortable enough with what’s happening to remain a part of it.
The paper provides additional concrete examples through transcript excerpts and quotes, and also explains the process by which the practice of collaborative inquiry evolves across a year. Really, it’s brilliant, and you should read it for yourself; my only intention in writing about it is to gather my thoughts 1) on the distinction between true collaboration and simply working in parallel alongside others, 2) on how teachers support the co-construction of understanding (what Staples terms “collaborative inquiry,” hence the title of the paper), and 3) on how important it is to establish common ground when doing so (something that had never really occurred to me before reading this and that I now can’t stop thinking about). I’ll stop now. Go read it.

Tuesday, December 16, 2014

conference notes part 2: what we're not talking about in mathematics instruction

Cultural competence. I normally don't like separating this from the "content" or "instruction" that I wrote about yesterday, because I think doing so reinforces a false dichotomy or a tacit acceptance that it's okay to be good at one but not the other. However, I've also found that when I talk about the two together, sometimes people hear what they want to hear-- either just the content/instruction part or just the cultural competence/equity part. I'm making the choice here to extract the latter not because I believe it can or should live separately, but to highlight and call attention to its importance.

One of the first presentations raised the question of what is "culturally and linguistically relevant to fifth graders," and described a research project in which a diverse group of students were asked to write their own word problems (it's unclear whether this was the focus of the project or a question that was incidentally raised while studying something else; I think the latter). The first example was about shark attacks-- and we were asked, rhetorically, to guess whether the author was a boy or a girl. The next few examples were about "bratty teenage girls," phone bills, and boyfriends, with the explicit statements that the "girl work is much neater, of course" and "we expected that their ethnic identities would come up, but they didn't-- not in a single one." Let's set aside for a second whether student-created word problems are truly a reflection of what matters to students or simply a reflection of what they think is "fair game" for word problems. Let's consider first, the assumed acceptance of gender norms (and not just gender norms, but heteronormativity, as one audience member thankfully pointed out; the presenter responded with an anecdote about getting relationship problems into textbooks and then shrugged, "there's clearly an adolescent boy culture and an adolescent girl culture"). Let's then consider the wink-wink implication that because no student wrote a word problem about his/her ethnic identity, that ethnic identity doesn't matter to students as much as shark attacks and cell phones. I don't know about you, but I can think of a hundred alternate explanations, all of which take offense at the assumption that just because a student is not White, s/he will necessarily (or should) write about his/her non-Whiteness in math class.

Later, when we talked about the skills we wanted our teachers needed to learn and/or attend to, and about building enough trust with teachers so that they could engage vulnerably as learners, we didn't talk about cultural competence even though the facilitators explicitly asked us to think about diversity and equity. I did manage to bring this up-- both because if we're to engage with what I hope are more and more teachers of color than have traditionally come through teacher prep programs, we as teacher educators must demonstrate cultural competence, and because if we're to support teachers who'll be working with students of color, we have to be able to model and explicitly talk about cultural competence-- but the conversation immediately shifted to curriculum and assessments in a not-even-remotely-connected way. Easier to talk about what other people need to do than to talk about what we ourselves need to do, I suppose.

Why do I write this? Because it feels really uncomfortable to be in a group of mostly White teacher educators/leaders (of the 80+ invited attendees, not including organizers, I counted 11 who were not White; give a little margin of error for folks whose racial identities are not visible, and that's still a much Whiter room than spaces I usually find myself in professionally) talking about mostly White teachers teaching mostly non-White students. I suppose some of the traditional teacher prep programs might primarily prepare teachers for predominantly White communities, but many of them do not, and most of the alternative programs in the room work specifically in urban and low-income communities which we know are mostly communities of color. And despite repeated prompting from the facilitators, the conversation never got beyond "all students should have access to learning," which reeks to me a bit of "let's make sure students of color, students with disabilities, English language learners, students in low-income communities, other traditionally marginalized students, etc. can get the same things that rich White students get (which is the default, of course; let's make Whiteness accessible to everyone)."

Sometimes, I think I'd rather not talk about it at all than see buzzwords dropped in this way-- culturally and linguistically relevant, equity, etc.-- and treated fairly superficially, as if we've checked off a box of "how to care about the things we're supposed to care about." Or should I be glad the buzzwords appear at all? And what is my responsibility to be able to talk about the things I do care about with more depth or clarity or force?

I don't feel like I'm in a position to call anyone out, especially because so many of the people in the room were/are figurative "lions" in the field (although I could also make the argument that it's even more important to do so then). For those of you who are more vocal, I'd love to hear how you think about when and how, and how that's shifted over the course of your career. In some professional circles I've developed a bit of a reputation for saying difficult things, but in many groups-- e.g., a roomful of mostly strangers who have been doing this work longer than I've been alive-- my comfort still matters more to me than standing up for what I believe are the important conversations, and I'm still working through how to change that.

Monday, December 15, 2014

conference notes part 1: how we talk about mathematics instruction

Last week, I attended a convening of teacher preparation programs-- both traditional, university-based programs and alternative programs-- who are thinking about innovation in supporting math teachers in the Common Core era. I've got a number of thoughts floating around:

This meeting confirmed that I would listen to Deborah Ball talk about dustmites if I could-- no matter what she's saying, she's such a skillful teacher and facilitator that simply being in the same room with her makes me feel like I'm learning how to be better. A few points that she either made directly, underscored from others, or implied/sparked (to me, anyway) about the language we use when we talk about teaching really stuck with me, because I think it's so easy to accidentally communicate something I don't intend to communicate, especially when using shorthand. Some of these I hope will be helpful for others' thinking; some are just reminders to myself about saying what I mean without glossing over important distinctions just to have a catchy or memorable phrase.
  • Explicit teaching is not the same as direct instruction: I think the citation on the slide for this was Engelmann & Colvin (2006), but I haven't found the original source yet. Teachers do need to make clear what students should be focusing on, what they should be thinking about, what they should be taking away, because otherwise, students are left adrift in the sea with no resolution (or thrown off the high dive… pick your watery metaphor). There are many ways to do that without giving students a model to mimic, which is what direct instruction-- the general idea, not the explicit strategy known as DI-- does.
    • Scaffolding is not the same as simplifying: Supporting a student to gain access to a task isn't the same as making the task easier or taking away the cognitive demand; if there's a jar of cookies on top of the refrigerator, scaffolding would be providing a stepladder or a stool, and simplifying would be moving the jar of cookies down to the counter. This may seem obvious, but I say because it sets up the next point:
    • Unpacking is not the same as breaking down: When we "break down" content, we identify its discrete parts, dissect it, and present our students with disembodied and decontextualized fragments. Think of breaking down boxes (and therefore rendering them unusable for their original purpose) or breaking down a campsite or breaking down a completed jigsaw puzzle. When we "unpack" content, however, we open it up to be examined. We make it accessible by pointing out interesting features or structures. Think of unzipping a suitcase and exposing its contents, or maybe taking some of the clothing out, so someone can construct their own outfit. Or of pulling back the curtain on a stage, and letting the audience's eyes rest on various props and pieces to figure out what setting and mood is being conveyed.
  • Students "doing the thinking" or "doing the heavy lifting" doesn't mean the teacher isn't: All the time, I hear (sometimes from my own mouth): "the teacher is doing all the work" or "students should be doing the bulk of the thinking, not the teacher." I think it's intended to mean "students need to be doing more thinking" or "students should be engaged in more intellectual work," but if we aren't careful, it can imply that the teacher should be doing less work, or less thinking. In fact, supporting students to engage in intellectual work requires an incredibly intense amount of thinking and work from the teacher-- to set students up to be successful, to elicit and listen to their ideas, to probe and challenge and build-- as anyone who's ever watched an "inquiry" activity crash and burn can attest. To put it another way, being less helpful doesn't mean doing less work.
    • Exploration isn't more important than discussion: Takahashi's 2008 presentation "Beyond Show and Tell" makes the observation that American lessons-- even when they're centered on a rich mathematical task-- often end after students have arrived at a solution. In a Japanese lesson, however, that's just the beginning; the bulk of the lesson is spent in neriage: discussing student solutions, finding similarities and differences among them, and building a collective understanding. @jybuell and I chatted a bit about this in terms of our own evolution as teachers-- I thought that getting students to figure something out was pretty cool, and it never occurred to me that there was more and more important work to do after that. He figured it out after a few years.
    • Focusing on the process doesn't mean the answer isn't important: Duh. I don't think any math teacher would argue that it's okay if students get wrong answers all the time as long as they engage fully in the process of reasoning (… and how valid can their reasoning be if it leads them to wrong answers?), but sometimes, I hear (and say) "the reasoning is what's important" or "it matters most how they got there" or "it's not really about the answer." All untrue. Just because we're now placing a greater emphasis on process and reasoning and flexibility and multiple methods doesn't mean we're lessening our attention to correctness. 
More tomorrow.

Wednesday, December 10, 2014

always sometimes never

I recently asked the #MTBoS who's written about Always Sometimes Never, and did y'all ever respond! Here's what I did with the resources you shared-- this text will eventually be part of an online course for novice math teachers, which hopefully explains the tone and awkward formatting (that's for the tech team to figure out). Thank you for sharing your practice with me and with the hundreds of teachers who will take this course in the next year!

In an “Always Sometimes Never” activity, students work with their peers to classify a set of mathematical statements as always true, sometimes true, or never true, and record their reasoning in some way. Determining and justifying whether something is true allows students to methodically generate and consider alternate hypotheses, which develops the habit of examining the full set of possible options. You’ve probably seen “Always Sometimes Never” statements before—especially if you’re a geometry teacher! It’s easy to simply give students these statements and let them wrestle, but structuring the lesson using the guidance in this learning experience will allow you to strategically maximize opportunities to elicit student reasoning.

Gray, Kristin. (2014). Always, Sometimes, Never… Year 2. Retrieved from

"Always Sometimes Never" can be adapted for a variety of different content topics, which makes it a generalizable instructional activity that you’ll be able to use throughout the year. Like "Talking Points" and "Comparing Quantities," it allows students to construct arguments and explain their reasoning. Because it is especially conducive to the use of case examples, however,  "Always Sometimes Never" further allows students to consider the role that counterexamples play in constructing arguments and the role that exhaustion plays in proof (i.e., have we considered all possible cases?).

[My hope here was to reference the video of Ms. Warburton teaching the MAP lesson Sorting Equations and Identities from the Teaching Channel, but they've put it behind a paywall :( The linked pdf lesson plan, however, is the document referenced in the next few paragraphs.]

When your students are working on this card sort, you may want to consider:
  • Are students able to make arguments (a claim supported by reasoning), or are they simply classifying the cards based on intuition?
  • Are students substituting numbers into the statements to test their truth? If so, ask how they might generalize without testing every single possible number.
  • Are students drawing conclusions after testing one or two cases? If so, ask them how many examples (or counterexamples) they need to find to feel confident that the statement is always, sometimes, or never true. This question will help students start thinking about the role of proof by exhaustion.
  • Are some groups finishing before others? If so, ask them to generate additional statements for each category. Specifically, ask whether they can generate statements with no solutions, one solution, two solutions, and infinite solutions.
Depending on students’ familiarity with “Always Sometimes Never” activities, you may want to consider one of the following options prior to the lesson:
  • Give students an individual task (as homework or as an exit ticket several days prior) to assess their initial level of comfort reasoning through “Always Sometimes Never” statements. See page 10 of the lesson plan for an example. Review student responses, using the guidance on page 4, and determine whether you might want to ask some of the probing questions to the whole class or teach a short mini-lesson prior to introducing the “Always  Sometimes Never” card sort. If you do, ensure that your lesson doesn’t teach students steps or processes for reasoning through the statements because doing so would significantly reduce the cognitive demand of the task.
  • Support the whole class in thinking through an example together prior to introducing the card sort. You could use the example on page 12 of the lesson plan (initially, simply post the algebraic statement; save the area models to use during the discussion), or use one of the examples from the card sort itself. Ask students to independently develop a conjecture (you could use a similar process as the Launch of the “Comparing Quantities” activity you learned about earlier) and then share their reasoning with first a partner and then the whole class.
After the card sort, choose a subset of statements to discuss as a whole class. You won’t need to—and probably shouldn’t—engage in detailed discussion of every statement. To choose the statements most worthy of class time, consider:
  • Which statements were sorted into different categories by different pairs/groups?
  • Which statements did students sort without fully fleshing out their reasoning, perhaps basing their decisions solely on intuition?
  • Which statements might have been sorted into incorrect categories due to algebraic or other procedural mistakes, rather than due to a misunderstanding of what makes a statement always, sometimes, or never true?
  • Which statements lend themselves to being justified via graph, diagram, or other visual representation in addition to algebraic manipulation?
  • Which statements might raise the question of how many counterexamples are necessary to feel confident in a given classification?  
To see how other teachers use “Always Sometimes Never” in their classrooms, and to see the specific statements and student materials they use, explore the following blog posts:
  • Fawn Nguyen shares links to additional statement sets and photographs of student work.
  • Tracy Zager has compiled these “Always Sometimes Never” statements for elementary-school students, which can be used as review or warm-ups in secondary classrooms.
  • Kristin Gray (click here for part 2) shares examples of student reasoning from her fifth grade classroom.
  • Lisa Bejarano offers an audio clip of her students discussing “Always Sometimes Never” statements.
A group of teachers known as the MathTwitterBlogosphere (#MTBoS) has constructed a living document with a range of “Always Sometimes Never” statements across grade-levels. Click here to access this document and identify statements that you can use in your classroom.

Additionally, Kate Nowak offers a variation on “Always Sometimes Never” in her Geometry classroom by asking students to draw counterexamples—pictorial or diagram-based arguments—to determine whether statements are true or false. This post explains how she does so, including a conversation with students about the role of faith, and provides an extensive set of classroom-ready statements.

Both "Always Sometimes Never" and drawing counterexamples, however, raise an important question: how do we know how many counterexamples we need? If we find one counterexample to a statement, we definitively know that hte statement isn't always true. Is that enough, though, to show that the statement's inverse is true, or to let us know how to revise the statement? That's what you'll explore in the next learning experience.

Tuesday, December 2, 2014


This post is me explaining something I currently believe, and also tracing the evolution of my development as a teacher/teacher educator. It's also a little bit all over the place, but I'm hoping that publishing something, anything, is better than waiting until I have the wherewithal to edit.

I used to think that anticipating student misconceptions and then pre-empting or eliciting-in-order-to-address them was one of the most important things teachers needed to know how to do. It required content knowledge, pedagogical knowledge, knowledge of students, etc., and helped teachers move students from point A to point B. In fact, I wrote a whole post about how to anticipate and respond to student misconceptions. The most important part of that post, however, was @blaw0013's challenge to me in the comments (and patience in supporting my thinking): what exactly was I calling a misconception? I encourage you to read the comments here, if not necessarily the original post.

I'm glad I found that comment discussion, because I've been contemplating a post explaining why the word "misconception" so raises my hackles, but couldn't find the genesis of this discomfort. In retrospect, I assume that Brian sparked a train of reading and thinking that included Benny (see @MathEdnet's summary here if you're not familiar with this classic Erlwanger article), various articles from the science education literature (and of course, the also-classic "Karen in the dark" video, which is linked in the aforementioned outdated post), and these two blog posts from Ben Blum-Smith, one on partial understanding and one on treating students' "I don't get it" as "I don't buy it." Sadly, I've lost track of much of what I've read in this realm, so if there's a seminal article or thinker I'm not citing here, please do let me know (terrible short-term memory relating to sources: this is why I would fail grad school). *Update: basically, just read this, from @brianwfrank, and all its contained links, because everything I'm about to say is essentially parroted from him.

Briefly put, I worry about the language of "misconceptions" because it's thrown around so lightly, to describe so many different types of thinking that we might consider incorrect-- careless mistakes, not-yet-knowing, conflating different ideas or procedures, incomplete understandings, etc. (here's one way I quickly explained some of these differences to a colleague a few years ago; not comprehensive, and I'd probably make changes if creating that document now, but sharing in case it gives someone something to push back on). I worry even more that focusing on "misconceptions" implies deficits in our students that only the teacher can fix or implies that the conclusions being drawn are illogical, nonsensical, or sloppy, rather than being constructed rationally as an extension of what the learner already knows.

I do still believe that it's important to spend time investigating and analyzing students' mistakes and errors, and that's part of what I find so useful about the conversations that @mpershan generates over at I've also found, however, that focusing novice teacher conversations on what students do know, and the partial understandings they are demonstrating, 1) allows teachers to see and talk about their students as thinkers and sensemakers rather than as empty vessels, 2) allows teachers to recognize how much value is already in their students' ideas, and 3) helps teachers much more quickly and constructively think through possible responses. For example, I've recently been using this student work sample in PD sessions, and I hear a lot of "wow, my initial reaction was just that this was wrong, but I'm realizing that this student actually understands so much!"

And when teachers respond intellectually to incorrect student work in this way, I think it influences the ways they respond socially and emotionally (when using the above sample with teacher coaches, I sometimes then shift the conversation to how this student's teacher responded, and what orientations or dispositions towards learning, towards mathematics, towards her role we can infer about the teacher from her response).

I'll expound on that soon, because I've been meaning to write about a related paper that blew my mind. Please hold.