Great question! We’ve uploaded two curriculum samples here: https://meld.sdsu.edu/index.php/curriculum/. These include the lesson plan, student handouts, additional materials, and a short video describing the goal and flow of the lesson. You’ll notice that in these lessons we have incorporated many of the Math Language Routines from the paper by you and your colleagues from Understanding Language, “Principles for the Design of Mathematics Curricula: Promoting Language and Content Development.” We found that adding these routines were very helpful to support student engagement in discussions as we designed the lessons. (And while you are at the project website, check out the cool icons that Kevin designed for the MLRs!)

Thanks for this question, Ed. In this project we developed our design principles and research instruments based on prior research. For the instructional design principles, we drew on research on student learning in the domain of linear functions (e.g., the work of Joanne Lobato). Focusing in your question, for the discourse moves, we have used more content-general tools such as the math talk moves in Chapin, O’Connor, & Anderson’s book “Classroom Discussions in Math.” For routines and structures specific to building on the assets of multilingual students, we have used work by Driscoll, Nikulah, & DePiper (“Mathematical thinking and communication: access for English learners”), as well as the “Math Language Routines” from Zwiers et al. at Understanding Language. This is not a comprehensive list, but a few of the resources we have used. 

You didn’t ask, but for what it is worth, for the analysis, we used methods of classroom discourse analysis, starting with the work of Courtney Cazden as a launching point. We also drew on Judit Moschkovich’s Academic Literacy in Mathematics framework, which is more specific to multilingual learners and focuses on the interdependence of mathematical discourse, practices, and proficiency.

Throughout all aspects of the project we draw on research and expertise from scholars in math education, including our advisory board members Judit Moschkovich, Kathryn Chval, Roschelle Gutierrez, Al Cuoco, Richard Barwell, and Joanne Lobato. 

Your question makes us think that one useful resource we can add to our website is a bibliography of resources for others who are doing similar work. Thanks for your question!

David, thanks for your question. We have definitely noticed this, and it’s an ongoing tension. We know from research that students learn more when they delve deeper into a few topics rather than superficially cover many topics. Nonetheless, this is a challenge that the teachers we worked with face all the time. If you have a solution, we would love to hear it. 

This is a big problem with our project:  the tension between the prescribed curriculum and the excitement and frustration that teachers who have learned things slightly "off topic" in our PD activities experience when they want to transfer that excitement and new ideas to their students.  We work with grade 5--an age that is so important in future career choices of students.  I tend to urge "create the best, most exciting learning opportunity for the teachers" while others insist "whatever it is, we have to stick to the prescribed curriculum."  Tension, indeed.

Michael, thank you for sharing this information with us. We agree with you that gestures are very important, especially in linguistically diverse settings. As part of our design, we incorporated multi-representational (e.g., equations, graphs, etc.) and multi-modal forms of communication (e.g., drawing pictures, manipulating representations, talking in whole-class and small group discussions, acting out mathematical contexts/ideas, etc.). While we allowed for gestures to emerge as forms of communication, we would like to learn how to systematically integrate them into our design of the lessons. What are some gestures you suggest we start with, and what are some ways we can begin to incorporate them explicitly into our lessons?

Hi Steven! Thank you for your question highlighting the resources of multilingual students. Throughout our project, we have worked to keep our focus on students’ resources, and we have found that multilingual students can enhance class discussions. We particularly connected our designs to prior research work on group work, funds of knowledge, and translanguaging. 

For example, discussing a meaningful problem in a small group setting provides low-stakes, collaborative, and personalized spaces, where students can share the resources they have. In such settings, students negotiate meanings together (e.g., “constant rate” and “speed”). This is why, in our redesigned lessons, we emphasized allowing sufficient time for group discussion before whole class discussions. 

We also discovered that the students bring in some surprising resources. In one lesson we saw a young woman on her phone throughout class. We thought she was off task, but later as we watched the tape, we discovered she was using Google translate to access the tasks that we had not translated. This made us more aware of, and open to, student resources!

Finally, in a linguistically diverse classroom, students can discuss how to say things differently in different languages, as well as what is the same. For one specific example, during meetings with our teacher collaborators (who are also multilingual), we had a rich discussion of the translation of “speed” to “velocidad.” While doing the translation, it led to interesting discussions of the distinction between speed / velocity in English and velocidad / rapidez in Spanish.

Hi Kristen-Thanks for your question!  We used a number of specific approaches to develop student participation in discussions. A few specific things we did were: 

1) Using open ended group discussion problems as an entry point. For example, the opening clip shows students discussing a “which one doesn’t belong” problem. This routine prompts students to attend to the similarities and differences across four images. Since there is no one correct response, this affords opportunities to focus on students’ reasoning / justification for “which one doesn’t belong.”

2) Related to #1, we planned lessons to alternate between small group and whole class discussions. We found small group discussions were critical for students to be able to use their full suite of linguistic resources, including multiple languages, to engage in discussion. 

3) Finally, we used “Math Language Routines” throughout our lessons to build a routine of sharing and talking. (Incidentally, one of the authors of the MLR paper commented below! Kristen meet Vinci, Vinci meet Kristen :)

As for dissemination, we have done several PD sessions locally and nationally, but we have not yet done anything for pre-service teachers specifically. That would be a great avenue for future dissemination. If you have suggestions for particular venues, we’d love to hear!

Thank you for your question. We’ll focus on the measures we used. We didn’t have much space to explain in this video, so we appreciate you asking! As one way to measure the effects of the design research for this video (and the paper we have in preparation), we did a deep analysis of classroom talk. We focused on one lesson that was taught before the intervention, redesigned, and then taught again. The topic of the lesson was reasoning about linear functions and average rate of change. 

We transcribed the classroom talk in all three instances where the lesson was taught. Then, for the analysis of classroom talk, we focused on whole-class discussions because not all of the small group talk was audible. We measured the responsiveness of teacher moves by coding the extent to which the teacher’s turns built upon or responded to a student’s idea. This is a coding scheme we adopted from Jessica Pierson Bishop’s research. A No/Low-Responsive move would be something like ignoring a student question or answer, or something like an affirmation or negation of a student answer. This level is typical in traditional patterns of classroom talk where the teacher asks a question, a student answers, and the teacher evaluates the answer. A Medium-Responsive move would be one that seems to be responsive to students’ ideas, but actually focuses on the teacher’s idea. This would include not simply evaluating a student’s idea but also giving the correct answer, as well as asking a student to fill in simple facts like units. A High I Responsive move would engage with a student’s idea but displays teacher reasoning, such as providing a counterexample to a student’s idea. Finally, a High II Responsive move would involve not only engaging with a student idea but also displaying student reasoning (e.g., asking a student to elaborate or explain more) or even change the direction of the whole class discussion to follow up on the student’s idea (“Wow, so Adri used a weighted average to get the average rate of change. Let’s make sense of what Adri did there. Can you come show us?”).