Classroom Interactions in a Common Core World
I’ve spent this year trying to teach a genuine Common Core Algebra I curriculum to high school freshman (my first time doing either of those) and I keep trying to find a way to write about with my experiences, but it’s hard not to get lost in all of the moving pieces that’ve happened this year. As the year wind downs, I guess the biggest thing I feel is: the Common Core shift is real and I feel it and the demands of the standards are affecting the types of interactions I have in my classroom. This post is about the challenges of encouraging genuine student-centered discourse in my classroom (and, specifically, the unexpected challenge of getting students to listen to each other), then sharing a strategy I tried this year that I think worked pretty well.
Looking at the wording of the Common Core standards and watching the architects talk about them, I’m convinced that genuinely teaching these standards requires a classroom where students are (1) talking to each other, (2) listening to each other, and (3) able to communicate their ideas in writing. I see this in the level of independence and self-startedness that the Common Core demands in their modeling standards and, if I want my students to be successful with these, then I need students to see themselves as bringing something valuable to a conversation instead of relying solely on the teacher as the one-and-only-knower of all information. I see this in how the standards are written – verbs like ‘Discover’ and ‘Understand’ are peppered throughout the standards, both of which encourage communication and debate and explanation rather than answering exercises. I see this in some of the released PARCC sample items (this question has always stood out in my mind) and, in Arizona on our AZMerit exam, in the emphasis on questions coming from levels 2 and 3 on the Depth of Knowledge chart.
This is tricky because none of these actions – talking, listening, writing – directly involve me, the teacher: my students need to be doing these and, more importantly, with each other rather than just with me. And yet, all of these indirectly involve me because I need to provide the opportunities for these to happen, which means I need to be very intentional about the types of questions I ask, how I elicit responses from my students, and how I provide feedback to encourage these behaviors. Intentionally trying to get students to talk to each other has become embedded in my practice – I’ve read strategies, seen resources from other teachers (the blogotwittersphere is especially strong in strategies for fostering communication), and I’ve seen this modeled enough times that I think I have a pretty good handle on trying to get students talking to each other and to me. The listening piece, however, has been a new and unique challenge I wasn’t expecting, but I think I’ve convinced myself that its the most important – listening, evaluating, and responding the what someone else says in a meaningful and productive way. Man that’s tough to teach freshman to do.
Example: I was at a conference this week where this pattern was put on the board: “3, 2, 5, 4, 7, 6, …” and the audience was asked what the next number was (it’s 9). We were then asked to share how we got our answers, and these strategies were then shared with the group. Some people saw the sequence in pairs – (3, 2), (5, 4), (7, 6) – and therefore the next pair is (9, 8); some people saw it as two sequences smushed together (like a piecewise sequence) – 3, 5, 7, 9, … with 2, 4, 6, 8, …. smushed between the numbers; some people saw a numerical pattern – subtract 1, add 3; some people saw that each consecutive pair summed to a pattern of its own (3+2 = 5; 2+5=7; 5+4=9) and used that to generate the next number (6+?=15; ? = 9) [personal note: I found this strategy to be the most interesting and a new playground for generating sequences]. All of these are useful and insightful strategies. Most of us in the audience, being interested teachers and respectful adults, listened to the other strategies and thought “oh that’s clever” or “ehh – mine’s better”.
But – here’s how this experience might play out in my classroom: students take a moment to get their answers independently, they share with each other to see how their answers compare, and then I call on people. My students, having their answer, are confident that they can respond should I call on them – I imagine subconsciously they’re thinking “Okay – I have an answer if he calls on me, so My Work Is Done“. As I call around the room, they listen half-heartedly and evaluate how similar each response is to their own only because I end every conversation with “thumbs up if this answer was similar to the one you discussed with your partner”. At the end of this discussion, we’ve discussed about 3-4 different ways to approach this problem, but students are still latched onto their way to solve the problem in case I pull a surprise attack and still decide to call on them.
This leads to the challenge: my experience has been that students aren’t even able to repeat back these other responses if I asked them, much less understand what they mean and see their value. More so, my students don’t see the reason why they need to be intellectually responsible for responses different from their own. To them, they got an answer and could explain it, so their cognitive work is done – listening to different strategies and evaluating their reasonableness isn’t something they’re used to doing. This presents a challenge to me because maybe the goal of this activity was to introduce the idea of piecewise sequences and so the ‘two smushed sequences’ strategy is really important for us to talk about. And as I try to transition to the next part of the lesson, I’ve left behind everyone in the class who didn’t use that strategy and wasn’t listening when their peers presented it, but still feels satisfied that they did what I wanted them to do because they had their own answer and could explain it.
So, this has been my guiding question as I keep finding myself in these situations and trying to react: how do I encourage students to have a reason to really listen to their peers thinking and use that thinking as the lesson continues? How do I create a culture of ‘student-as-giver-of-new-information” instead of “teacher-is-only-person-who-can-give-new-information”?
Quick sidenote: it would be easy for me to add a teacher move saying “these are all great strategies, but the one we’re really going to talk about is … Let me explain it again to make sure everyone understands it before moving forward“, but there are some subtleties to this (especially the italicized part) that I have worries about. First, it reinforces that the teacher is still the most important person in terms of giving new information and, if I do this often enough, students will realize that they can always wait until the end of the discussion to listen to my little summary. Also, if I do this too often, it implies that there is a ‘correct answer’ that I’m looking for in this type of activity, which means my discussion of different strategies isn’t motivated by a genuine interest of how students solved the problem but by my search for the ‘correct strategy’ that we’re really going to talk about. I’m a big believer that once students have even the smallest suspicion of this deception, their motivation to take risks drops to 0 and their engagement bottoms out as they sit there wondering “when is he just going to tell us what he wants to talk about?”.
Other End Of the Spectrum: If I don’t do this teacher move – adding some teacher-guided explicitness to a specific strategy – then a lot of my students will struggle with the next activity/problem/task if they weren’t listening and evaluating different strategies (for example: if my next sequence was “1, 20, 4, 18, 7, 16, 10, 14”, the students who missed the ‘smushed sequence’ strategy would struggle much more than I want). This last year, I feel like I spent a lot of classroom time regrouping, re-eliciting the responses, then sending students back to work on their task. This takes time. I want this to take less time.
Anyway – this has been part of where my head has been – finding some good teacher moves to handle to situations above. I’ve been looking for resources and strategies to get students to listen to each other, and trying different things that I’ve invented in my own class. Here’s what I’ve tried so far (although I don’t think I’m anywhere near an expert yet):
Weekly Mathematics Discussion Notetaker
This was something I created to try and get students to listen to each other as we had class discussions. I expected it to be a purely extrinsic motivation (a grade), but it had some unintended consequences that led to some intrinsic motivations (more on that below). Anyway – students got it at the beginning of the week and turned it in at the end of the week. They were responsible for 5 quotes from other students where that student explained their thinking. This could be full-group or small-group. They couldn’t use themselves and they couldn’t use the teacher. Eventually, I tried (but failed (but will try again next year)) to add that they can’t use the same person twice during the week, hoping to encourage more people participating in discussions. This became a weekly grade and was one of the few things they couldn’t turn in late. I encouraged them to look for because statements and to think of this as recording notes that they could look at later, so the notes needed to be clear and complete. When I graded it, I didn’t look for correctness of the statements (so “The y-coordinate is 3 because the y-coordinate comes first” would still get full points) because that’s not the purpose of the assignment and, especially during some class discussions, incorrect things are said that are clarified later and I didn’t want to penalize students for not knowing the difference yet. I did, however, provide written feedback to clarify any incorrect quotes. This became a part of my students day-to-day operation in my class – they had their pencil, paper, and the discussion recorder. The back of the handout was a weekly planner because I thought that would be a good use of the space on the back, especially for my freshmen – it didn’t factor into their grade.
Some Observations: Overall, I’d call this a success with some really nice intended and unintended consequences. I noticed students paying more attention to class discussions which was the whole point of this. But, more than that, I noticed students being more conscious about phrasing their answers in complete sentences and using the word ‘because’ in their responses. In fact, this led to a new type of classroom interaction: I’d ask a student to answer a question and explain their answer; their answer is incomplete or mathematically imprecise (“what’s the better mathy word for that horizontal line on the coordinate plane?”); I keep pushing them until I’ve gotten a complete, precise response from the student but with lots of interruptions from me; AND THEN (here’s where the magic happens): I ask them to put it all in a complete sentence using the word ‘because’, which triggers the other students to get ready to write it down, and now I’ve got a student saying a complete, mathematically precise explanation and the rest of the class really paying attention to what they’re saying and how they’re saying it. But someone doesn’t hear it, so I make another student restate it, and now two people have said the complete, mathematically precise, student-generated explanation. Yes – this takes time, but I think it’s been worth it.
More consequences, most of them unintended: students got a sense of pride from being someone whose name gets written down on everyone’s paper as contributing a good response (next year, I think I want to make a wall of ‘good quotes’ to keep track of good responses through the year). My students who liked to answer questions and get called on started to get frustrated because: if they’re always answering questions, and they can’t write their own name down, then their discussion notetaker never gets filled out. So, these same very-active students started encouraging other people to interact in class so they could fill out their notetaker – it was a nice unintended student-driven reaction to the need for these to be filled out. Students also started holding each other accountable for more detailed responses – for example, let’s say I wanted a student to do a problem up at the board and talk through it. Maybe the discussion goes like this:
Student: “I did this. Then this. Then this. Then I got this and that’s the answer”
Me: “Did he/she say anything that you could write in your discussion notetaker?”
Me: “Are there any questions you’d like to ask him/her so you can add something to your discussion notetaker?”
(Students starts asking why and can you explain questions to the student presenter)
(Or: no one asks anything and the opportunity is gone)
I’d be remiss if I didn’t at least mention: I had to adjust a few things after the first few weeks as I saw lots of people trying to copy responses on the Friday due-date. I had a conversation with the classes re-emphasizing the point (to reward those who share during class discussions and who are actively paying attention), I explained that I would make adjustments for absences so you didn’t feel pressured to complete the whole thing just because you were absent, and I said it was okay to copy a response if it just happened and you missed the tail end of it. Then I laid out some consequences and it wasn’t really an issue anymore.
So – in addition to this whole discussion notetaker thing, I’ve also been looking at…
Resource: 5 Practices of Orchestrating Productive Mathematics Discussions. This is a pretty stellar book and I like it as a starting point for how to orchestrate tasks and activities that really require students to generate different strategies. It’s also got some great advice for how to respond to students so the teacher still sits in this role of facilitator and guide rather than taking on the role of information-giver. I don’t really know what else to say – if generating productive discourse is going to be a focus in my classroom, this book is important.
Resource: MAP Project PD Modules on Questioning and Student Collaboration. The PDFs linked in bold have some really great questioning and interaction strategies, which I think is half the battle. Students can only have genuine conversations if I ask the right type of questions and encourage the discussion in just the right way. Students will see themselves as independent learners once I learn how to interact with them in a way where I’m out of the way yet still holding them responsible and pushing them forward. I haven’t looked at these in-depth, but I think I need to.
Reflection: I don’t want to write a ton on this because it would be easy to dovetail into a curriculum conversation, but I guess I wanted to mention: as I think about the topics/lessons/days that have generated lots of discussion versus the ones that didn’t, I tend to think that the truly rich discussion days happen when the topic lends itself to connections between multiple representations – specifically, connecting to some kind of real-world context. Teaching parabolas as connected to different scenarios & contexts led to explanations that weren’t arbitrary or rule-based but were instead interpretations tied to the scenario that we were investigating. The same thing happened when I introduced equations to represent parabolas – having graphs and equations and scenarios available as resources for students to share their answers led to deeper explanations and discussions than if we were examining each topic separately. On the opposite end of the spectrum were my lessons on factoring and properties of numbers (commutative, associative, etc). I didn’t connect factoring to anything else other than a procedure (admittedly a bad teacher move), so the explanations during those weeks were really weak and un-insightful. The properties of numbers were very rule-based and definition-based, so it was hard to get explanations that were anything other than a restatement of the rule or definition. These units need work; units that work in multiple representations seem to be an easier playground for meaningful explanations and discussions.
Conclusion: I don’t really know if I have one, but I’m excited to start the discussion notetaker at the beginning of next year, I’m hoping to connect as many topics as I can to a real-world context or multiple representations, and I’ll keep trying to shift the role of information-provider away from me and onto my students.