emergent math

I struggle with how “special” we treat math in schools. It’s not uncommon for math teachers and departments to run professional development apart from all other subjects. Or use different classroom norms. Or instruct entirely differently. Or blog or tweet exclusively about math. Math teachers have their own software, their own language, different and separate from the rest of the school.

The entire design of New Tech Network schools, for whom I work as a Math and School Development Coach, purports that schools work best when staff practices, protocols, norms, and buy-in are all aligned. For example, entire staffs are committed to the same norms, the same school culture, and the same protocols. So I’m often biased toward thinking about how kids are experiencing school, rather than just math.

Still, even with that alleged common understanding, I’ve heard so many times from non-math teachers, “oh, that’s the math department, we just let them do their own thing.” Or, from math teachers, “oh, this is math, we do it differently in here.”

It’s also quite rational behavior for teachers. If I have limited time for planning and reflection (if any), I’m not going to use it to explain what we’re going over to teachers who won’t give precise feedback. And If I’m employing instructional software as my teaching tool, my peers have literally nothing to offer me.

Is this OK? Is this best for a student’s schooling experience?

I often wonder how are students experiencing and witnessing this. Do they see the disconnect between math departments and the rest of the staff, like the way children intuit when parents aren’t getting along? Do they experience firsthand the pedagogical isolation of a math class compared to the (more often) aligned approaches of other subjects?

It makes me wonder if there’s an upper limit to how great a school – or even a math department – can be if they’re so often sequestered from the rest of the staff.

I worry sometimes that perhaps a great folly of all these rich math resources online is that they can allow math teachers to remove themselves from school norms and ways of being (editor’s note: I want to emphasize the word “can” here, not that it does, but that they can. Are we cool, now? Cool.) Great tasks and lessons are the technical solutions. A staff coming together to determine how to best support students is an adaptive one. (For more an the technical vs. adaptive terminology, head over here.)

So how can math teachers engage productively with the rest of the staff?

At the most successful schools I work with, there are a few common threads, which I wrote about at the beginning of the school year. I would like to highlight/reiterate the notion of, as a staff or grade-level department, examining samples of student work across subject areas. In addition, consider intentionally inviting peers into your classroom. Facilitating a really cool task sometime in the next couple weeks? Send an invitation and/or record yourself so others can see what you’re doing, and perhaps by proxy, get oriented to what fun math can actually be! Follow it up by inviting yourself in to a non-math classroom to observe and learn.

To be sure, the discipline of math is peculiar, to the point of being existential. There are certain teaching moves that are special to the discipline. There are attitudes within students unique to the subject developed over time that we must examine and treat. And to be fair, every subject has their own discipline-specific ideas and norms. Math classrooms just seem to take it to the extreme in terms of looking different.

This is the tension I find myself in: just how different is math as a subject? When is it appropriate and beneficial to think about math-specific teaching strategies vs. general strategies? To what degree does math’s “specialness” hinder or help the overall goal of a school where students and adults feel connected and successful?

I’m not sure I have any concrete answers to these questions, but I do know a couple things: that teaching math is enhanced by leaning upon fellow awesome math teachers and their lessons AND students experience school best when overall teaching practices and norms are aligned.

I’m chatting with the Global Math Department on 12/15/15 about using student work as the driver of teacher learning. Consider this post a repository for pertinent links, my slide deck, and a comment section for further conversation.

GMD_LASW

Link to working google doc for the session:

[Global Math Department 12/15]

LISTEN: Podcast in which we with Belleville New Tech facilitators about using Looking at Student Work (LASW) to drive teacher learning: [School Innovations Podcast Episode 305: Looking at Student Work, Driving Instruction]

WATCH: Video of a staff using a protocol to analyze student work

More on Looking at Student Work (NTN Website)

A quick Would You Rather today.

I was inspired by a teacher asking students to model airport parking as a means to get at linear equations. I thought I’d frame it similarly, but using the WYR format, because I like the WYR format.

I live a good distance away from a major airport, and I travel quite a bit for a living so I think about this all the time. I have to balance a flat, $80 fee (for both ways) vs. paying for parking (per day, the big variable here), gas, and tolls.

Enjoy!

It’s a lot easier to complain that students don’t know, say, their multiplication tables than to actually teach multiplication.

Setting aside the oft problematic mindset of a teacher complaining about what “these kids” don’t know for the time being, consider actually teaching to the gaps you feel are present.

Let’s be clear: not a single one of us have entered the school year 100% satisfied with where 100% of our students are at math-wise. There’s always something that was allegedly “covered” in previous years and for whatever reason was not retained by the students. A couple years ago I was at a relatively well-off suburban school where 99% of their students graduate and go on to college and even those teachers were complaining about what their students did and didn’t know.

Often under the guise of unspecific complaints about students “not knowing their basic math facts” or “numeracy”, teachers sometimes pass blame upon students, The Calculator, or their prior schooling. What I often don’t see happen is addressing those gaps in knowledge. Sometimes a cursory remediation worksheet is handed out, and after-school tutoring is offered, but many times I don’t see teachers actually teach to those gaps. Y’know: teaching kids these lugubrious “basic math facts.” Even more specific complaints about how students “don’t know how to do fractions” (whatever “do” means) are ripe for teaching opportunities, rather than tsk-tsk-ing.

Which is unfortunate because there’s never been a more robust cache of resources to remediate in a healthy, fun way. It reminds me of that bit from Arrested Development where Lucille Bluth brushes off her poor raising of Buster because “kids don’t come with a handbook.”

[Ron Howard voice] In fact, there are countless books that address the very learning gap you’re complaining about. NCTM has so many publications that would probably be perfect. Or go here and click on the grade lower than you. Shoot, just go to amazon and type in what you feel your students are struggling with.

If you feel your students lagging in a particular area of their learning, I’d suggest rather than complaining and sending them to a worksheet or instructional video, consider doing some learning yourself and find a book, blog, text, paper, resource, or teacher to teach you how to teach to this area. I’ve learned so much from my non-grade level colleagues about teaching number sense, rounding, fractions, ratios / proportions, and even an alleged area of expertise of mine: algebra (thanks Andrew!).

I mean, if you need a specific recommendation, I’d consider learning how to facilitate some number talks via Elham and Allison’s Intentional Talk and go from there. Or follow ’em on twitter and get their awesome advice for free! (But seriously, get their book.)

Also, we’re not talking about shutting everything else down classroom-wise, lest you’re worried about losing precious class time. While coverage is overrated, let’s put that aside for now, shall we? We’re talking 10-20 minute activities and discussion here, maybe a couple times a week. Stop complaining and start learning how to teach this stuff. If we want students to learn, we probably ought to do some learning ourselves, no?

Besides, teaching these skills and concepts is fun. This has probably been my biggest takeaway of the year so far: leading number talks is so fun, I’d do it even if I wasn’t addressing learning gaps.

“I lecture, but I do it in a dynamic, interactive way!”

Teachers are sometimes justifiably defensive about their lectures. In many circles, lectures are a four-letter word. This flies against not only hundreds of years of pedagogical practice, but cuts against the teacher-as-expert model of instruction.

Of course, there are good reasons for that too. Lectures can be too long and a not terribly good medium for conveying information. I do think it is a good medium to generate excitement and momentum about a topic, as well as brief explanations of the pitfalls in a particular concept.

This isn’t a pro- vs. anti-lecture argument. This is just a brief atlas of four different lecture models, with animated .gifs of course.

Lecture Model 1: Teacher talks

This is our lecture hall model. The lecturer talks aloud, leading listeners through a topic.

It’s not as bad as it sounds though, potentially. This is basically our TED Talk model and people love TED Talks. Provided the material and/or the presenter are interesting enough, this is a perfectly fine model of information and interest conveyance. If you can pull it off and retain interest, more power to you.

Lecture Model 2: Teacher talks –> Teacher questions a student –> Student responds –> Teacher talks

This is actually the lecture model that inspired this post. When teachers claim their lectures are “interactive”, a lot of times this is what they mean. I’m not sure this is terribly interactive. It can be! But it also can not be.

This is probably my least favorite type of lecture, and probably the most prevalent. Under the guise of a “dynamic, interactive” lecture, it often becomes a “gotcha” lecture and “can you recite what I just said?” lecture. It’s basically the tactic I use when my kids start to tune me out.

Certainly there are benefits to such a lecture, but unless you’re being incredibly precise about it, it’s quite possible for a student to go the entire time without engaging with the topic at hand. I suppose the option is usually on the table: “students are free to ask questions in the middle of my instruction!” But in practice, unless that’s truly a norm the lecturer often goes unquestioned.

It’s also inefficient. Consider a 30-minute lecture in which only one person is talking at a time (the student or the teacher). That’s 30 minutes of “discourse.” Even in Lecture format one (outward), you’d have 15 minutes of outward lecture and then 15 extra minutes to do something to get other kids talking. This lecture could easily have been replicated via Zaption or other interactive videoing program.

Lecture Model 3: Teacher talks –> Teacher questions a student –> Student responds –> Student responds

This is what a “dynamic, interactive” lecture ought to look like. This is what I think teachers are going for when they go down the path of Lecture model 2; it’s more difficult to pull off, takes more time, and allows enough grey area that some teachers aren’t willing to cede.

I’m talking about the teaching move where a student responds to a question about something and instead of the teacher confirming, denying, or expanding on the response, a fellow student responds. The teacher sets the conversation in motion but the students become the primary questioners and conversationalists. We see this modeled well in literary circles, but not often in math class.

To encourage this dynamic, a teacher may prompt “Jane, what do you think about Jack’s idea?”, “What words in Jim’s statement resonated with you?”, or, as Kate adeptly incants, “Hey, so-and-so, would you explain your understanding of Bianca’s solution?” Some classrooms have gotten so good at this that the prompt is no longer needed. In Brette’s classroom it became a norm that students would routinely ask things like “can you prove that to me?” and “how are you certain about that?” Music to any lecturers ears.

Lecture Model 4: Teacher talks –> Prompts students to talk –> Students talk

Not unlike Lecture Model 3, in this case a teacher prompts additional thinking. The teacher may talk for about 30% of the lecture time. There is a lot of “turn and talk…”, and “tell your elbow-partner…”, and “in your groups, discuss…” and so on. At this point, a teacher becomes untethered to the front of the room and can join in the small conversations, prompting additional questions or deeper thinking.

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Next time you’re lecturing, I encourage you to think about the animated .gifs and consider diagramming the conversation, replete with arrows and dots. Researchers do this regularly. Consider becoming a researcher in your own class by tracking the conversation. Invite a peer in to do it for you or video yourself and watch it back. If you’re going to lean on the lecture as a tried and true means of conveying critical content, at least make sure you spend some time diagnosing the lecture model itself.

The Golden Circle

It’s become almost de rigueur to “Start With The Why” or at least to share that title track from the Simon Sinek TED Talk playlist. I often use it to structure my talks or professional developments. I don’t necessarily disagree with the mantra. However, in practice, I seem to often get more out of teachers only once they implement the “What.”

The start of the school year has seen a lot of the “What” thus far. The “What” in the teachers I’ve had the pleasure working with, has fed the “why”, not the other way around. That is, when a teacher facilitates a 3-Act Task, or a Would You Rather?, or a Which One Doesn’t Belong, or a Desmos Activity, or PBL or PrBL Unit, their practice changes as a result of the succeeding reflection. Jo Boaler’s Low Floor / High Ceiling tasks actually start to find and bubble up the cracks in student learning (or in teacher practice). Starting with the “why” feels almost like getting teachers to reflect on something that hasn’t happened yet, or at least the way I clumsily package it. Perhaps even more often, a teacher may be falsely reassured that he/she is attending to the Why if they don’t ever offer a compelling What. A teacher that assigns solely rote problems may never expose the gaps in his/her Why: I want kids to be good problem solvers, and just look at all these problems they’re solving!

I wonder if that’s in part due to the fact that most of us are kind of aligned with the “Why.” We all want students who are persistent problem solvers and like math. That’s true of basically 100% of teachers. You don’t need to build buy-in around that “Why.” The part where we differ and where teachers have had disparate training experiences is in the “How” and “What.”

One of the more successful coaching experiences I had last year was with a teacher I never had the chance to “pre-brief” the Why. She was hired early in the year, but after school had started. And she took the resources and started trying stuff out. It was through the reflection of those facilitation experiences that she grew as an educator. We never talked about the Why around inquiry or student engagement or anything like that. At the end of the year, she was rocking and rolling with Problem Based Learning, using many of the tasks that had been provided and developing some of her own. Early this school year I’ve had the pleasure of seeing teachers implementing a lot of “Whats” and the reflection and feedback has pointed to the Why, rather than the other way around. And when things go poorly we can talk about that as well. Oftentimes it’s when the tasks break that teachers reconsider the inner sections of the Golden Circle (the How and Why).

Similarly, the most influential PD activities I’ve participated in or facilitated are those that involve some sort of student work analysis. Once we have a What, we can start to pull apart how it points to (or doesn’t) our Why. Or only then do we start to develop our Why. I love Math Mistakes because the tasks and student work provided elicits conversation about the purpose of mathematics and how we teach it.

Does mindset change practice or does practice change mindset? Or does practice change practice?

Of course it’s possible that I’m using the lens of my own predisposition to break and try new things. I didn’t really start reconsidering my practice and how I did things until I tried something quite new and foreign to what I’d been doing before. Only after I made a huge mess did I start thinking more deeply about my craft, including the Why. While I’ll probably continue Starting With The Why ™ for my own planning and reflection, I’ll probably ease up on my emphasis about its importance.