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Centering class discussions around student mathematical thinking has been identified as one of the critical components of teaching that engages students in justifying and generalizing. This report shares analysis from a larger project aimed at describing and quantifying student and teacher components of productive classrooms at a fine-grain level. We share results from this analysis of 39 mathematics lessons with a focus working with public records of students’ mathematical thinking. 

The development of a model that explains how teachers learn from teaching is critical for informing the design of quality professional development, which in turn can support teachers’ effectiveness and student learning. This article reports the authors’ effort to develop a model that brings together critical findings from existing research to unpack when and under what conditions teachers learn from teaching. Grounded in evidence drawn from research relating to teachers’ learning and practice, the authors build a rationale for the Learning from Teaching (LFT) model, introduce each component of the model and propose two conditions that increase the likelihood of teachers’ learning from their own teaching. 

In this report, we discuss five forms of reasoning about multiple quantities that sixth-grade students exhibited as they examined mathematical relationships within the context of science. Specifically, students exhibited forms of sequential, transitive, dependent, and independent multivariational reasoning as well as relational reasoning. We use data from whole-class design experiments with students to illustrate examples of each of these forms of reasoning. 

Teachers in the elementary grades often teach all subjects and are expected to have appropriate content knowledge of a wide range of disciplines. Current recommendations suggest teachers should integrate multiple disciplines into the same lesson, for instance, when teaching integrated STEM lessons. Although there are many similarities between STEM fields, there are also epistemological differences to be understood by students and teachers. This study investigated teachers’ beliefs about teaching mathematics and science using argumentation and the epistemological and contextual factors that may have influenced these beliefs. Teachers’ beliefs about different epistemological underpinnings of mathematics and science, along with contextual constraints, led to different beliefs and intentions for practice with respect to argumentation in these disciplines. The contextual constraint of testing and the amount of curriculum the teachers perceived as essential focused more attention on the teaching of mathematics, which could be seen as benefiting student learning of mathematics. On the other hand, the perception of science as involving wonder, curiosity, and inherently positive and interesting ideas may lead to the creation of a more positive learning environment for the teaching of science. These questions remain open and need to be studied further: What are the consequences of perceiving argumentation in mathematics as limited to concepts already well-understood? Can integrating the teaching of mathematics and science lead to more exploratory and inquiry-based teaching of mathematical ideas alongside scientific ones? 

Collective Argumentation Learning and Coding (CALC) is a project focused on providing teachers with strategies to engage students in collective argumentation in mathematics, science, and coding. Collective argumentation can be characterized by any instance where multiple people (teachers and students) work together to establish a claim and provide evidence to support it (Conner et al., 2014b). Collective argumentation is an effective approach for promoting critical and higher order thinking and supporting students’ ability to articulate and justify claims. The goal of the CALC project is to help elementary school teachers extend the use of collective argumentation from teaching mathematics and science to teaching coding. Doing so increases the probability that teachers will integrate coding in regular classroom instruction, making it accessible to all students. This project highlighted Gloria (pseudonym), a fourth-grade teacher from Cohort 1 because of the extent to which she went from fear of coding to fluent implementation. Initially, Gloria was comfortable engaging her students in argumentation, explaining they already used it in mathematics with Cognitively Guided Instruction (CGI). However, she was “terrified” about learning to code because she didn’t view herself as proficient with technology. She was willing to overcome her fear of coding because she saw the value in providing her students with coding experiences that would help them develop the necessary skills for our increasingly technological society. In the course of three months, Gloria’s instruction progressed from using simple coding activities to more sophisticated coding platforms. This progression in her coding instruction paralleled the change in her personal feelings about coding as she moved from “terrified” to “comfortable with it”. 

Much of the research on the development of professional noticing expertise has focused on prospective teachers. We contend that we must investigate practicing teachers as well, and in particular practicing secondary teachers, because they bring with them years of teaching experience and are situated in unique contexts. Hence we studied the longitudinal growth of the professional- noticing expertise of a group of practicing secondary teachers (N=10) as they progressed through a 5-year professional development (PD) about being responsive to students’ mathematical thinking. Results indicated that the first half of the PD supported their interpreting and deciding-how-to- respond skills, and the second half of the PD supported their attending skills, which were already strong even before the PD. We compare these results with the activities that occurred in the PD and discuss implications for future research and PD programs. 

In our project, we develop curricular materials to support prospective secondary teachers’ development of MKT and provide professional development (PD) opportunities for instructors ho will teach with these materials. In this paper, we examine the ways in which mathematics faculty engage in the teaching rehearsal debriefs included in the PD to answer the question: To what instructional interactions do instructors of mathematics content courses attend during rehearsal debriefs enacted in PD? Findings show that mathematics instructors attend to all types of interactions but attention is influenced by instructors’ mathematical knowledge.