Category Archives: TMC

Instructional Routines, Teacher Thinking, Student Thinking

I spent my morning sessions at Twitter Math Camp working with David Wees, Kaitlin Ruggiero, and Jasper DeAntonio on the Contemplate then Calculate instructional routine: experiencing it, unpacking it, and rehearsing it as teachers. I’m excited to put it into action this year, and I learned a ton in the sessions both about the routine itself and the benefits of the thinking that goes into it. A quick summary of Contemplate then Calculate:

The routine starts with a mathematical task. Here are three that David, Kait and Jasper modeled with on the first day:

Screenshot 2016-07-24 at 3.48.30 PMScreenshot 2016-07-24 at 3.48.46 PMScreenshot 2016-07-24 at 3.48.59 PM

The task is launched by clarifying the goal of the routine — to look for shortcuts or strategies using mathematical structure. The image is flashed and then hidden, without a question, for about a second. Then, students get a chance to talk and and share what they noticed that they think is mathematically important. The image goes back up, clarifying the problem if necessary, and students try to calculate in their heads, using whatever strategies they can find, often with the help of the noticings. Students then get a chance to share strategies with each other, and the teacher selects and sequences strategies to share and annotate for the whole class. Finally, the class engages in a meta-reflection, thinking about how the structure used in different shortcuts could be useful  in other contexts.

That’s a pretty high-level overview; there’s a ton more subtlety to it that’s worth exploring. Resources from the morning sessions are here, and this document has links to videos and other great resources to learn about Contemplate then Calculate.

I loved experiencing the routine as both a learner and a teacher, and I think it gets at some mathematical habits of mind that I don’t do well in my class. But what I’m particularly fascinated with is what went into planning to teach a Contemplate then Calculate routine.

During the session I worked with Sadie Estrella, planning to co-teach the routine. There are tasks on the New Visions website for Algebra I through Algebra II, and each includes slides and other relevant information. It took us about 15 minutes to work through a planning template to prepare to teach the routine; I’m sure this would get faster if I did it more often. And, since the routine doesn’t require any handouts and the slides were made, we were ready to go.

This is a pretty significant contrast with my usual lesson planning time, which is largely spent creating the handouts I’m going to give to students, fighting with formatting, spending only a fraction of my planning time thinking about what I’m going to do and even less on how I’m going to do it.

David ended the last session with these slides, which were particularly striking to me.

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This gets at a big reason why I’m excited about this routine. During our planning, we were thinking almost exclusively about student thinking — about what strategies students were likely to use, what mathematics in those strategies was worth highlighting, and where the task fits into the broader scope of the mathematics students would learn. I learned a ton just from those minutes I spent preparing to teach the routine. And while I’m likely to be generating my own tasks for some classes this year, the fact that so much of the routine is set will allow me to continue focusing as much of my thinking as possible on student thinking and how I am going to move that thinking forward. That’s a pretty cool thing.

Many of us will be using the #CthenC hashtag to share about this routine over the next year. If you’re interested, follow along, or give it a shot yourself and share how it went!

TMC16, Race, and What We’re Not Talking About

Twitter Math Camp was awesome. Julie just wrote a great guide to catching up if you weren’t there. It was inspiring to reconnect with so many incredible educators, and it has left me even more excited for the coming school year. Lots to think about, lots to process. But of all the ideas that I have spinning around in my head right now, I keep coming back to Jose Vilson’s keynote (click through for the video).

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“So why did I choose race? Well, I mean, look at me. And then look at y’all. Then look at me again… I had to choose race because that was the one space I didn’t think we talked about enough.”

Jose talked about how, as math teachers, we ask critical questions, we prepare for teachable moments, we expect non-closure, we stand on inquiry and openness, and we allow for multiple pathways. We do all of those things to help our students learn math. To become critical race thinkers, we need to apply those same types of thinking to our conversations and our actions around race.

I don’t write often about issues of equity or racial justice, and Jose’s talk was an important reminder that my silence is actually a pretty loud message. It’s time to change that.

It’s also important to look at our community as a whole. Two years ago at my first Twitter Math Camp, the three keynote speakers were white men who were not classroom teachers. They were all great speakers, but we also sent a message about the voices our community chooses to value. The last two years have featured three women and three men, voices of color and elementary voices, half classroom teachers. By my rough count, in 2014, male presenters outnumbered female presenters almost 3:2. This year there were more women presenting than men, though only barely. We had a number of sessions around social justice and equity, compared to (maybe) one two years ago.

This is progress. But it’s important not to mistake progress for success. We have a long way to go to get the voices we need into the conversations we’re having.  And this is an iterative process. We need to be more inclusive to improve our collective critical consciousness, and we need to be more critically conscious in order for more folks to feel included.

It’s worth noting that I helped to perpetuate the problem. When I was asked to give a keynote, I had a lot of ideas running through my head, and lots of questions. One question I didn’t ask was who the other two keynotes would be, and whether a range of perspectives would be represented. As a white man from a background of privilege, it’s something I need to say. And I didn’t say it.

I’m not sure where this goes next. Writing about racial justice and social justice is not my strong suit. And my voice and my perspective will always be limited. But these are conversations that we need have, because they’re essential to the work that we do. Lani Horn has a great metaphor here, borrowing from John Dewey (around 42:00 in the linked video). It’s easy to stay up in the realm of theory and not navigate the uncertainty of working in classrooms with complex power dynamics and students who are forming their identities. But it’s like teaching someone to swim without ever actually getting into the water. And this uncertainty is the water the teachers swim in, and the water that can make us drown. If we pretend it isn’t there, we’re ignoring the exact power dynamics that have created the problems we face today, and creating new challenges for tomorrow’s students.

This post doesn’t have a clever ending. I don’t have any answers. What I do have is a drive to follow the advice of Jose, of Rochelle Gutierrez, of Kaneka Turner, and turn it into action in my classroom and conversations in the community.

What If Everything I Think About Teaching Is Wrong

That’s what I was often thinking during six fascinating hours in Alex Overwijk and Mary Bourassa’s morning session at Twitter Math Camp.

They have dispensed with traditional units entirely, instead teaching content through activities. An activity could last anywhere from a single class to three weeks. Each major topic in a course comes up within the first few weeks (Al called this “unloading” standards), and is spiraled a number of times before the end. Here’s what the curriculum might look like:
Screenshot 2015-07-30 at 4.46.42 PM
Each column is an activity, and each row is a standard.

Important Notes:
Al and Mary use standards-based grading, which seems like a necessary component. Note the T1, T2, T3, T4 and SE columns — those are assessments, and the standards that are being assessed. They also start the year by co-creating, with students, criteria for great questions, and students decide in groups which questions are most worth answering. They are unafraid as teachers to steer the class toward specific questions, but allow some freedom for students as well. It’s definitely a slow process, building students’ problem posing skills, but a valuable one as well. An activity might start with an image like this:
sequoia

Look closely. No, more closely. Now what questions do you have?

Al and Mary showed us activities using ropes to illustrate linear equations, stacking cups in all kinds of formations, using a memory game to illustrate quadratic and exponential functions, and using different sized squares to illustrate linear functions, quadratic functions, and the Pythagorean Theorem. They were a ton of fun, and I learned a bit of new math along the way.

A huge part of their success is clearly the quality of activities they have prepared — but through our working group, we are working to build a bank of great activities. Check out what we have so far here.

Why Activities?
An activity-based classroom is ambitious teaching. Making this work requires a pretty significant collection of activities, careful organization to track standards, deliberate messaging to students, and flexibility to deal with the inevitable messiness of activity-based learning.

This approach also has some pretty enormous benefits. It matches what we know about brain science — that in order for learning to be durable, students need practice spaced over time and interleaved between different topics. The way most math classes are structured, practicing skills for a short period of time and then moving on, is practically designed to create the illusion of mastery — knowledge cycles through short-term memory without any incentive or reason to build the neural networks necessary for long-term retention. Activities are also more engaging for students, for obvious reasons. Al first tried this approach 8 years ago because he realized his lowest level 10th grade class wasn’t learning everything. They redesigned the course, with the goal that first it had to be engaging. If students weren’t learning, at least they wouldn’t leave the room hating math. They built this system from that starting point, and have had remarkable success.

One great question from another participant helped drive the point home for me. The participant asked: What if you plan an activity, and it doesn’t work — students aren’t asking the right questions, they aren’t ready for the math, or are focused on other details of the activity besides the relevant standards? My answer — while a failed lesson might look a bit uglier in an activity-based classroom, I teach plenty of failed lessons. Maybe kids are sitting quietly, but more times than I can count this year, a class left my room having learned nothing of substance. I struggle with that. If activity-based teaching has some pretty serious possible benefits, and my students and I might have a bit more fun in the process, it sounds like a good deal. Most of all, if it takes away some of the illusion of learning I get from massed practice and students cramming for a rest, I will be a much better teacher for it.

Where Am I Going From Here?
I’m not going to go full-bore with activity-based teaching this year. I’m starting at a new school, and have lots of things to figure out moving from 8th grade to high school with new curriculum, routines, and lots more. It also feels like a huge risk, and not one that I want to crash and burn with the humans I teach. But I want to incorporate these ideas where I can, and see if I can “test run” them to consider moving forward with this approach. I’ve been thinking about spiraling activities in the context of my Algebra-II class for the fall semester. It’s all juniors, and spans a pretty significant range of content. The topics I’ve been given for the fall are:
Expressions & Equations
Linear Equations
Quadratics
Exponential Functions
Functions (from basic function concepts through inverses and function transformations)
Exponents & Logarithms
Complex Numbers

Two approaches I’m considering, with the caveat that I haven’t started work yet, so I have no idea if these will fly:

No unit on expressions and equations. Instead, I spiral those topics throughout the semester with a variety of activities — I have a few ideas so far, and I’m sure I can build some more. They’re critical skills that my students need, and I think the spiraling approach could be a critical one. And activities focused on expressions and equations will naturally wrap in other standards, and provide a nice change of pace.

No unit on functions. Instead, spiral function concepts throughout the semester. I have a few ideas for this already, including Function Carnival, Graphing Stories, and more, as well as some reading I’ve been doing about how students understand functions and where the idea of a function came from anyway. This is a bit more ambitious, especially with respect to inverses and transformations, but I think if I can successfully wrap functions into everything else we do (with the possible exception of complex numbers) students will have a much broader and deeper understanding of the function concept.

Finally, I’d like to continue my approach to homework from last year. Distributed practice on a variety of topics, but without questions on that day’s learning goal. Students get continuous practice on key concepts throughout the semester, homework is less dependent on how well class goes, and I get a chance to review and address misconceptions from the whole course on a daily basis. It’s small, but I think it makes a big difference.

I’m going to spend some time over the next few weeks puzzling through what each of these might look like, and in particular if I feel like I have the depth and breadth of activities to do them right. My goal is to have a very clear idea of the abstract ideas I am pushing for through spiraled activities, the different perspectives students need to be able to look at them through, and a number of concrete classroom activities that will move students in that direction.

Professional Learning

I’ve been doing a lot of thinking the past week about exactly how it is that I get better at teaching through an experience like Twitter Math Camp. Much of this has been prompted by David Wees post titled “Learning at Conferences“. David’s suggestions boil down to finding one idea during the conference on it, and go to many sessions on that idea — that any single session is unlikely to make a significant change in your practice. He goes on to suggest that it is well worth the time to skip a few sessions and network with colleagues, real or virtual, at the conference in order to have time to reflect on these ideas.

I think David’s ideas are excellent — at NCTM in Boston this year, I fell into the trap of both trying a little bit of everything, and going to sessions that didn’t challenge my ideas (Dan Meyer or Steve Leinwand are fun, but it’s more of a hoo-rah for their ideas than a paradigm-shifting experience after having seen them a few times). While I made some great new connections and had a ton of fun at that conference, I’m skeptical I learned very much.

I do want to propose two alternative ways in which I think I have learned, in particular from Twitter Math Camp, that don’t fit into what David is talking about. These might only be possible because Twitter Math Camp is a uniquely excellent conference experience, but they’re ideas I want to consider relative to all of my professional learning — blogs, Twitter, a PLC at my school, and anything else.

Building a Vision of Great Teaching
I don’t see great teaching very often. I am not a great teacher. I don’t mean to be a skeptic, but let’s acknowledge that most teaching is not great teaching. And not in some negative all-teachers-are-tenured-and-lazy way, just that we’re all figuring this thing out and it’s a tough place to reach. And it’s often hard to articulate what great teaching looks like. There’s plenty of disagreement on the topic. I was in a school where for awhile I was the only teacher interested in number talks and three-acts, among other things. I was figuring those things out. Over time, what’s happening in the other classrooms down the hall becomes normal, and I’m missing out on opportunities to improve my teaching because I don’t know what would be useful for me to learn.

This was the biggest impact of Twitter Math Camp for me. I came away with some ideas to use in my class, though I’m not sure how well I will implement them. But more than that, I came away with an enormous vision of great teaching — what those teachers do and how they think. I owe more thanks than I can keep track of. From each keynote, to my morning group, to the afternoon sessions, the barbecue and late night and early morning conversations at the hotel, I was lucky to have the ideas and insights of many brilliant teachers in my ear. Sam Shah wrote this morning about how the landscape of math teachers on the internet has changed over the last 8 years, and how different people have developed their own “brand” — being known for their expertise in a specific area. Well, I had a ton of expertise, in all different areas, that I am lucky to be able to have listened to the last few days. Each one pushed me to think in new ways, and kept me hungry to keep getting better at this crazy job we’re all trying to do.

This is a subtle process. There are no huge leaps forward, few immediate changes I will be making because of what I have experienced. But these interactions, over time, will build the teacher I am becoming. I feel this particularly as a young teacher, too often painfully aware of my failings. But if I stay cooped up at my school, in my room, I can convince myself that what I’m doing is alright. I don’t have that vision. And I sell myself, and my students short. So, to every single person I talked to — helped to build my vision of great teaching this week. Thank you.

Getting Better Through Reflection
I try to get better by reflecting on my teaching, but let’s be honest, that’s hard. It takes time, intellectual effort, and a certain expertise to be able to identify what the changes are that I want to make. The second big thing I took away from Twitter Math Camp wasn’t about something new I was going to do, it was about some new ways of thinking about teaching. I heard Alex Overwijk and Mary Bourassa share a completely different approach to teaching, how they went about it, and why they think it works. They gave me a totally new set of tools for thinking about what is working, and what isn’t working, in my teaching. I will be hearing their voices this year when I realize a week after a unit ends that my students have forgotten everything and won’t get a chance to see it again. Ok, I know, I spent six hours in their morning session — but this was a big shift for me. I don’t know if it will cause me to make a change right away, but it gives me a completely new perspective that I am excited to apply.

Lani Horn had a similar influence with her keynote. She spoke about teachers’ professional learning, and her lens shed light on the ways that teacher conversations about teaching affect their learning. I wrote more about my takeaways from her session here — but in just an hour, she gave me three new tools for considering how my conceptions of my teaching interact with my learning about teaching. I can take those ideas and use them to be more reflective this year, and further my growth.

What I’m really talking about here is a paradigm shift. When can a new perspective, or a new framework, or a new lens for looking at teaching change the way I interpret what is happening in my classroom, and help me get better through reflection outside the limitations of that conference? Twitter Math Camp has a uniquely dense lineup of folks who helped me to do this thinking, and it is one of the most important ideas I am taking away.

These are very different perspectives than I took at my first conference, now almost a year and a half ago. Then I was desperately searching for new things — something I could do in class that would work a little better. Those things still exist, and I am hoping to continue accumulating them. But most of my learning happens in the grind of the day to day, and these last few days have made a huge difference in my ability to take that daily grind, pick out what is most important, and know the changes I need to make to get a little bit better tomorrow.

Thank You, Twitter Math Camp

I am sitting in the airport about to leave Twitter Math Camp, and I can’t stop thinking about everything that I experienced the last few days. I’m trying to capture as many ideas as I can with the goal of using it to be a little bit better in the classroom this year, but I’m also happy knowing that what I got out of TMC goes way beyond any new activity or question or explanation I use this year.

It’s the community. Or, in the words of Lisa Henry, “it’s the community, stupid”. It’s the conversations — talking about questioning during a session, or designing tasks late at night, or polynomials at the barbecue, or the intricacies of our students early in the morning, or the messiness of teaching at the airport when I thought I was done. It’s the teachers who inspire me to be more creative, more compassionate, and more thoughtful in the classroom. It’s the vision of what great teaching looks like that is pieced together from keynotes, morning sessions, anecdotes, jokes, and heart to hearts at the piano bar. It’s what reminds me why I do this work, why it is so important, why it is worth all the time and effort, and why I will keep doing it for a very long time.

I will try to put together some more thoughts the next few days, but right now I need to get on a plane, and I feel so happy with everything I have experienced the last few days and what it means for my practice as I say goodbye to a wonderful summer of math and teaching, and say hello to a new school and new adventures, and to a year where I will continue to lean on those wiser than I am.

Robert Kaplinsky: Questioning

Robert Kaplinsky ran a great session yesterday at TMC on asking questions to see what students actually understand. His title was: Improve Your Questioning Skills to Formatively Assess Student Understanding. We took turns going through a really fun role-playing protocol, where a teacher tried to probe a student’s understanding to see what they are actually thinking. There was great discussion about different types of questions that do a great job of probing understanding and asking students to elaborate on their thinking, rather than just saying yes or no. I was lucky to be partnered with Alex Overwijk and Christopher Danielson, and I ended up with a great new tool for getting students talking about their thinking.

So a student has solved a problem, and I suspect that they don’t understand what they’re doing — they got it wrong, or maybe they got it right for the wrong reason. I often end up going dwn this rabbit hole with the student. Even if I ask great questions, they tend to 1) realize they got the question wrong and 2) not want to relive it and answer all my questions. Each of us used a similar move where, when we realized there was something about the student’s thinking that didn’t work, we gave the student a short task to do to reveal that thinking, rather than continuing to probe and dig through the evidence we already have.

Examples:
In the first scenario, a student draws this picture of 1/3:
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You can make a pretty good guess at the misconception to probe student thinking, and Al started with a few quick questions asking Christopher to explain his thinking. He then asked Christopher to draw a new picture of what 1/3 might look like. Christopher drew this:
Screenshot 2015-07-25 at 7.09.56 AM
That reveals a lot more thinking, and also puts some more of the thinking on the student. We didn’t get into what to do next — the goal was to see where the student was coming from — and I loved that choice of a questoin.

In the second scenario, I was asked to order the decimals 0.52, 0.714, and 0.3. I ordered them as:
0.3, 0.52, 0.714.
Christopher tried to dig into my thinking, and I was very confident I was right. He then asked me to order two new numbers: 0.15 and 0.30. I now had:
0.3, 0.15, 0.30, 0.52, 0.714.
That forced me to do some mathematical thinking, and pretty clearly revealed my misconception. He didn’t stop there, though, asking me to write some new numbers that were larger than 0.714. I stuck with my student personality and gave him 0.715, 0.716 and 0.900. I loved this approach — rather than picking an argument with the student over the answers they already had — and were very confident about — getting some new information to reveal that thinking.

In the third scenario, Alex needed to find the median of the set of numbers
3, 7, 4, 2, 9
Alex said that the median was 4. I asked him a bit about his thinking, then, copying their moves the first two round, gave him a new set to find the median of:
2, 7, 3, 4, 9
Alex picked 3. Golden.

I don’t know what to call this move, but I really like it. It worked great for us, and I think would work even better with a student. I hate ending up in the situation of doing a “post-mortem”, forcing a student to relive a question, and this technique gets just as much or more information, but shifts the student’s attention to a new task.

There was lots more to Robert’s session that I’m skimming over, but this was my big takeaway, and it’s something really concrete that I hope to put into practice within the next few weeks.

Andrew Stadel on Mistakes

I went to Andrew Stadel’s afternoon session at TMC yesterday: Math Mistakes and Error Analysis: Diamonds in the Rough. I got out of it a fun technique to get students thinking, and some new perspective on building deep knowledge.

Andrew talked a great deal about the value of mistakes, and using those mistakes as opportunities for students to learn. His first proposal was simple, and looked like this:
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Simple, and an awesome way to get students thinking beyond answer-getting to the mathematical structure. I’m excited to give this a shot in my class this year.

Building off of Andrew’s ideas, I’ve been thinking a great deal about how students come to understand broad, abstract ideas. Dan Willingham has been a source of great thinking here. I’m coming more and more firmly behind the approach that students build abstract understanding through the variety of examples of an idea they encounter, and from looking at the connections between them — it’s less about finding that “perfect example” and more the sum of all the little things we do to get kids thinking. Building this takes time and hard intellectual work. While I was in Andrew’s session, I jotted down this idea on the back of his handout:
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This is inarticulate, obviously, but it was my brain trying to get out an illustration of the “pyramid of abstraction”. There’s this big idea of exponents. It starts with some simple relationships, which we like to codify into rules. It builds up through more complex relationships and different examples, and leads students (hopefully) to a broad and transferable understanding of exponents. I’m stealing from Jason Dyer’s ideas here “teaching to the negative space” — that we need to teach what a concept isn’t, as well as what it is. The pyramid of abstraction starts with simple relationships, and this includes both the positive and the negative. It reminds me of something Dylan Wiliam quoted during his conversation with folks at PCMI last week — “Our memories aren’t good enough to remember algorithms perfectly. We need to understand so we can do a kind of ‘in-flight repair'”. In the same way, as student understanding is starting to develop, it may not be enough to try to remember relationships that are true — humans forget. If students are also examining and remembering relationships that are false, it provides one more means to move their thinking forward and build strong, flexible conceptions of math.

There was a great deal more in Andrew’s talk that I am skipping over — he posted more resources on his site here and we had great conversations and looked at other techniques. But this little tidbit — one idea of what I want to bring to my classroom, and a move forward in my thinking about how students learn — is what I am taking away from his talk.