Hi Susan Jo:

Thank you so much for viewing our video! By inquiring responsively we mean posing questions to surface students' thinking, making comments and asking questions that are in harmony with that thinking, and posing challenges at the edge of students' thinking. As a researcher, I work from models of students' thinking that I am always in the process of developing further. So, in my research, responsive inquiry is a regular activity, and in the first two years of the project I studied my own teaching in after school design experiments with middle school students. Then in the later years we worked with teachers and, like you and many others in the field of math ed (e.g., Susan Empson and Vicki Jacobs' work on responsive teaching), we used student work from teachers' classrooms in professional development settings. For example, teachers and research team members formed a year-long study group to look at student work from teachers' own classrooms and developed questions to ask to surface, assess, and (potentially) advance students' thinking. But I should say this: Our project was not squarely on working with teachers, and in my view others in the field have done more with studying this important practice. What we found is that inquiring responsively is a critical feature of differentiating instruction--not that we know well how to support teachers to do so. I'd be curious about how you support it in your own work with teachers. My conundrum is that organizing one's ideas about students' thinking is an important piece of doing it well, and many teachers have little time for that.

Although we've examined some of what teachers do in supporting mathematical argument in the elementary grades (see chapter 2 of But Why Does It Work?), we haven't focused in on small group interactions, and I think this is such important territory.  I think your naming this work "inquiring responsively" is great--the words themselves imply listening to and connecting with students' ideas.  I was just looking at a piece of videotape from a second grade class in which the teacher is interacting with a pair of students who are trying to articulate a conjecture about equivalent addition expressions, and I noticed that what she does so skillfully is to stick close to what the student is saying in her responses, while also giving him some alternatives about how to continue, but without taking over his idea.  

oops--I meant Chapter 4

Yes, I think that describes it really well: I would describe the sticking close to students' ideas without taking over their ideas as "harmonizing" with their thinking. It's so easy in the course of teaching to push toward a particular direction and then completely miss what a student is saying. For example, in the very first experiment we did in this project, one of the 7th grade students commented that the multiplicative relationship stated in a problem was "approximate." He said it a few times in small group work, and then in the next session in the whole class. I did not really know what to make of the comment, and several other students disagreed with him (though some agreed), but I knew enough to listen to him and hear him out. It turned out that that comment was very informative in helping us understand not just his thinking but a whole group of students' thinking about multiplicative relationships between unknowns. And it led to design changes in the second experiment. So you can learn a lot from harmonizing. But it does require setting aside the pathway or pathways that you think you might be on with students.

That's a great word for this.

Hi Julie! Thanks for viewing our video. After working with Amy on this project for several years now, I think the thing I'm most excited about is the potential for this work to inform future middle school teachers in methods courses. The research team spent a lot of time this year selecting video clips and student work samples and writing targeted questions about them for use in the middle school methods course Amy teaches. We want PSTs to understand that differentiation is not just putting different tasks/numbers/etc. in front of different kids but rather that it is deeply intertwined with students' thinking. We hope that the materials we created help PSTs begin to consider ways to get to know their student thinking and use what they learn about students' thinking as they explore ways of differentiating. We're hoping to present or publish on these materials soon, but we're not yet sure what form that will take. 

Hi Julie: Just to chime in, we do have a practitioner article that we hope will come out in the next year. And we will be presenting some work at the NCTM regional conference in Salt Lake City in October (there is a strand specifically about differentiation there). And hopefully we will present some work at AMTE (proposals are due tomorrow!)

Thanks, Amy and Rebecca! I will watch for publications/presentations. I'll be at AMTE, so hope to attend there for sure!

Hi Denise! Thanks so much for viewing our video! Amy will likely have some things to add, but I can answer your question in regards to the final phase of the experiment, when Amy co-taught a unit with the classroom teacher. For both classes, we knew well ahead of time what unit Amy would be co-teaching, and we had access to curricular materials (both classes were teaching from CMP) from that unit. As a research team, we looked at current research in the field (for example, students' rational number knowledge) and, using that research, anticipated what types of thinking we might see. We planned tasks to elicit this thinking so that it would be more clear to us and also considered how we could differentiate for students who were thinking in different ways. So we were definitely thinking holistically - about the unit as a whole, and how students would think/grow across the unit. Yet, of course, we also had to consider individual lessons, so there was definitely a lot of day-to-day discussion, too. And it's important to note that there were several of us - a whole research team - working together to plan and support implementation of the lessons. A lot of the work was done behind the scenes by the research team - and one or two of us would meet with the classroom teacher to share our ideas and get her feedback/hear her concerns, then modify. We're definitely considering ways to make this type of teaching more accessible when there isn't a research team behind the scenes. 

What are your experiences working with teachers as a coach? Do you feel it's been more meaningful to focus on one lesson at a time ro to think more about big ideas within units? 

Both!  But unfortunately I dont always find the time to do this work with teachers as well as I wish we could.  Having a research team behind the scenes would certainly help :) 

Years ago our district found the time and resources to "unpack" modules (units) with grade level teachers led by instructional math coaches like myself and to this day teachers continue to tell me that our unpacking sessions were the best PD they received regarding teaching and learning mathematics.  We would start with taking a close look at the standards embedded in the unit and then move to assessment pieces found in the end of module assessment as well as state released test questions so we could see and feel what the end goal was.  We would do the math and share our solutions which would lead to discussions about visual models and conceptual understandings we needed to focus on throughout the unit.  We would then identify the most impactful lessons and tasks found in our curricular materials to lay down a thoughtful pathway to develop the skills and understandings we wanted our students to gain rather than blindly moving through our curricular materials lesson by lesson.  The process left teachers feeling empowered and confident in their teaching of the grade level standards.  Then at the school level, instructional coaches could work with individuals on a specific lessons in a similar way.  Pre-planning always included doing the math together to anticipate trouble spots or potential student misconceptions.  Together we could generate purposeful questions and plan to look for certain strategies or solutions to highlight in the closing discussion.  So to answer your question, I find both practices useful when working with teachers but do notice if I plan a lesson with a teacher who hasn't yet thought of the goals and big ideas of the unit, our conversations are not as rich as there were when we had unpacked the unit together.

Thanks for viewing the video Jessica ;-). The basis for providing choices is so deeply rooted in one's understanding of student thinking and big ideas in the domain, right? I think there is a lot to be done in understanding how teachers understand this aspect of differentiation.

In our work we have found that all teachers--even experienced ones or ones with traditional math backgrounds--need to work with identifying and understanding big ideas. In other words, identifying and understanding big ideas is itself an inquiry process--it's not cut and dried, even for those with math backgrounds. In our Teacher Study Group, teachers did work collaboratively to design tasks that provided choices for students, and we also met monthly to discuss work that their students produced on these tasks. That gave opportunities to talk about big ideas as well as student thinking together, which was nice.

And thanks for viewing, Beth! Do you have initiatives in your district for teachers to differentiate instruction?

Wow, small world! Hi Heidi! Director--of this showcase or CADRE?

Hi Lizchen--

Robin and Fetiye have already provided nice responses below, so I will just chime in with this: You will always have some times like this, where you are not sure how to respond. When I am not sure what to do, I try to listen harder, give space, and ask questions--and I always keep in the back of my mind that how I am thinking about the problem is just one way to think. Or, even if I have several ways of thinking about the problem based on having worked with students in the domain for awhile, I don't assume I know all the ways they might be thinking about it. I know that I can't challenge a student well, usually, if I don't have a good sense of how they are thinking.

Thanks, Amy, Robin, & Fetiye,

What you all said really hit on me. When I tried to elicit student thinking and make decisions about how to respond to their thinking. I felt like I was fighting with the other me, i.e., the teacher side of me. It’s true that we cannot anticipate every single piece of student thinking. Doing reflection helps better how to respond to students.

Also I want to resonate with Amy’s conundrum, i.e., the difficulty and importance of organizing one’s own ideas, from a teacher’s persepctive. Admittedly, teachers should hold back their understanding of the problems and respect students’ ways of thinking. Meanwhile, teachers are not responding to students on an emergent basis. Instead, teachers, I think, should have stored a list of content and pedagogical knowledge/questioning skills and be ready for the interaction with students.

Hi Lizhen,

Thanks for sharing your thoughts about the importance of teachers' content and pedagogical knowledge and questioning skills. I definitely agree with you on these, as they are helpful tools for teachers to learn about students' ways of thinking.

Hi Lizhen! 

I taught for quite a while before coming to grad school full time, and I also feel a big struggle between my teacher self and my researcher self! When I was teaching in my own classroom, every task, question, activity, etc. had an end goal - I wanted students to learn the content for whatever standard I was teaching. As a researcher, I'm not necessarily as concerned with the outcome as I am with getting inside kids' minds and trying to understand why they're doing what they're doing - how their thinking and understanding influences their actions and quantitative decisions. I did find, though, that reading the current research on students' ratio reasoning (esp. the Lobato, Ellis, Zbiek Essential Understandings book) and anticipating various students' responses in terms of their multiplicative concepts really helped me ask strategic questions that brought out their thinking in a way that I believe would have been less successful if I hadn't had that background knowledge. 

I second Pai below in thanking you for viewing ;-). I agree that differentiation could be used to frame your work because one can differentiate for many purposes. And yes, we are intertwining students' mathematical thinking and practices to differentiate in our project. I would love to learn more about how you are tailoring mathematical learning environments to students learning the language of instruction (probably should have posted this on your video page!)

Amy

Hi Amanda: This is such a great question, and it's nice that you are asking about families because that is an important factor in systemic change, which is partly cultural. I don't study this myself, but I know that there have been a few recent cases of detracking in California: For example, San Francisco Unified has detracked its high school math programs. I wish I could upload a document of information here; I'll try a link but don't know if you will have access: https://www.nctm.org/uploadedFiles/Standards_an...

I know that this work has required a serious community effort, with teachers and district leaders and parents working together. The National Council of Teachers of Mathematics has now come out with a call to detrack secondary math program (just last summer), so this may motivate more community efforts like this one.

That link worked- thank you. 

I only asked this question because you mentioned in your video the benefits of detracking, and training teachers to differentiate is critical, but so is getting the community on board. Of course, all classrooms benefit from differentiation, embracing learning differences, and challenging students appropriately! 

Oh, you are right to ask! it's just a very hard question, supporting and studying systemic change.

Hi Jomo! Thanks for watching our video and for asking your questions. I'm sure Amy will chime in with her thoughts, but I thought I'd weigh in, too. As far as patterns in students' mathematical understanding, we are still analyzing our data and finding those patterns. But in terms of supporting teachers' efforts to differentiate, I think the biggest thing we've found is that really listening to students was very important, as was having an understanding of the research on student thinking on the topic being studied (ratios, for example) so that we could anticipate possible responses and ask strategic questions. I personally found that as I was circulating and asking questions of students, I was able to probe their thinking more effectively when I knew the content well, when I had anticipated how those particular students might respond, and when I asked strategic questions to either confirm or complicate the responses I had anticipated of them. The biggest thing I've learned from this project, in terms of supporting teachers' efforts to differentiate instruction, is how absolutely essential it is to really get to know each student's thinking. The better you know students' thinking (rather than relying on your assumptions of their thinking), the more effective the differentiation will be. 

As far as your second question goes, I suspect out of school experiences have an influence on student thinking. This project, however, didn't collect any data on those experiences, so answering that question is outside of the scope of this particular work. 

Hi Jomo:

Thanks for viewing our video and for asking good questions. And thanks to Rebecca and Rob for their responses. One thing we have observed in students' mathematical understanding has to do with how they organize and nest units--I'm talking about discrete ones as well as measurement units. This has an impact on science as well, since you are often measuring and relating quantities to each other. We have a framework for thinking about how students organize and nest units that has been found to influence their mathematical thinking in a variety of domains. So, we used that as one source for differentiating instruction in our project, and it could be a source for teachers as well, since there are assessments teachers can make about this framework (see Developing Fractions Knowledge, a research-based book for teachers:
https://us.sagepub.com/en-us/nam/developing-fra...). I think developing ideas about a framework like this one can help teachers to differentiate instruction, but engaging in some sustained professional development/inquiry is essential.

Regarding out of school experiences, they certainly have an impact. But we were not studying those in this project. But that could certainly be a part of future work on learning how to differentiate instruction. One of the goals in differentiation is to endeavor to move from a "one size fits all" approach to seeing, respecting, and responding  to the variety of students' thinking (or other characteristics) in a classroom... my basic premise is that if that were happening more regularly, we would have more people thinking of themselves as mathematical thinkers--more people to feed into your STEM pipeline, in effect.

Hi Kristen:

Thanks so much for watching and for your comment! Absolutely yes, curricular materials play a critical role. In the first part of the project I taught groups of 9 students in after school design experiments and did not follow any particular curriculum (was exercising my freedom in that way). But in the last part of the project when I co-taught units with classroom teachers, we were using CMP3. I know you have a lot of connections to CMP3 ;-) and I very much enjoyed your video about the technological tools you are using with it now--looks very interesting and inviting for both students and teachers.

In our project, the teachers were in their first year of implementing CMP3 when we did our first whole classroom experiment. (And unfortunately in our state, IN, there is little support for professional development for teachers, so they had had very little PD. They have had a small amount now, and the teachers I work with are craving more of it.) Anyway, I was really excited at this confluence of events because I admire a lot of things about CMP, and it definitely supported our purposes in its focus on problem-based, discussion-rich mathematics instruction. One thing that we did not necessarily find in it was a wealth of options for supporting students who are different levels in their multiplicative reasoning. Students' multiplicative reasoning is one way I think about (or "operationalize") students' diverse ways of thinking, and we were designing for this in the experiments. Because I like CMP3 so much I was "dreaming" about writing some alternative pathways for students to engage with the big ideas in the units that would be tailored to students at different levels of multiplicative reasoning. (We did some of this in each of two units that we co-taught, but not broadly.)

So, to get back to the broader question, in my view curricular materials with some good alternative number choices and problem pathways would be a great resource for differentiating instruction. I do think that teachers would still need to do the work of getting to know student thinking, of course--that's the foundation for everything. If you have any interest in talking more about this, just let me know ;-).