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Abstract The complexity of mathematics teaching is especially evident in lessons where teachers build on students’ genuine ideas, such as problem-based lessons. To enhance teachers’ capacity for rich discussions in problem-based instruction, we have developed a unique approximation of practice: digital asynchronous simulations where teachers make subject-specific decisions for a virtual teacher avatar. The simulations are based on materials and principles from a practice-based professional development (PD) program, implemented with small groups of teachers. The self-paced simulation model offers flexibility and scalability, allowing more teachers to participate on their own schedules, but it lacks key affordances of collaborative PD. To examine how to leverage the affordances of collaborative, practice-based PD, this paper uses a design-based research approach to explicate the mechanisms in which digital simulations can support mathematics teachers’ learning about problem-based lessons. We focus on two cycles of design, implementation, analysis, and revisions of the simulation model, drawing on data from focus groups with mathematics teacher educators, prospective teachers’ performance, and teachers’ reflective assignments. The analysis illustrates how two design principles –Authenticity to the teacher’s work, andNuanced feedback– were transformed to better reflect aspects of practice-based teacher learning. We argue that self-paced, asynchronous simulations with indirect feedback can effectively emulate aspects of collaborative, practice-based PD in supporting teachers’ growth. The paper also contributes to the literature on mathematics teachers’ noticing and decision-making, examining how the two interact in simulated environments. We suggest implications for designing practice-based asynchronous digital simulations, drawing on emerging technologies. 

This paper contributes to understanding the work of teaching the university geometry courses that are taken by prospective secondary teachers. We ask what are the tensions that instructors need to manage as they plan and teach these courses. And we use these tensions to argue that mathematics instruction in geometry courses for secondary teachers includes complexities that go beyond those of other undergraduate mathematics courses–an argument that possibly applies to other mathematics courses for teachers. Building on the notion that the work of teaching involves managing tensions, and relying on interviews of 32 instructors, we characterize 5 tensions (content, experiences, students, instructor, and institutions) that instructors of geometry for teachers manage in their work. We interpret these tensions as emerging from a dialectic between two normative understandings of instruction in these courses, using the instructional triangle to represent these. 

This research article contributes to the growing literature highlighting the potential for innovation in mathematics education through design cycles that involve creative risk-taking and failure-based learning. Specifically, we explore how “failed” cycles of StoryCircles—a practice-based professional development approach that centers on teacher collaboration—have been productive in fostering innovations within the program. Our focus is on the challenges that arose in our efforts to enable feedback mechanisms within the StoryCircles system that support teachers’ interrogation of their own instructional practice, as they collaboratively develop lessons and expand their collective knowledge base for teaching mathematics. Through examples of three challenges, we illustrate how various lesson artifacts, including those constructed by teachers in anticipation of implementation and those extracted from actual implementations, failed to serve as the sole source of feedback for supporting teachers’ growth. 

Wolbachia is a widespread endosymbiont of insects and filarial nematodes that profoundly influences host biology. Wolbachia has also been reported in rhizosphere hosts, where its diversity and function remain poorly characterized. The discovery that plant-parasitic nematodes (PPNs) host Wolbachia strains with unknown roles is of interest evolutionarily, ecologically, and for agriculture as a potential target for developing new biological controls. The goal of this study was to screen communities for PPN endosymbionts and analyze genes and genomic patterns that might indicate their role. Genome assemblies revealed 1 out of 16 sampled sites had nematode communities hosting a Wolbachia strain, designated w Tex, that has highly diverged as one of the early supergroup L strains. Genome features, gene repertoires, and absence of known genes for cytoplasmic incompatibility, riboflavin, biotin, and other biosynthetic functions placed w Tex between mutualist C + D strains and reproductive parasite A + B strains. Functional terms enriched in group L included protoporphyrinogen IX, thiamine, lysine, fatty acid, and cellular amino acid biosynthesis, while dN/dS analysis suggested the strongest purifying selection on arginine and lysine metabolism, and vitamin B6, heme, and zinc ion binding, suggesting these as candidate roles in PPN Wolbachia . Higher dN/dS pathways between group L, w Pni from aphids, w Fol from springtails, and w CfeT from cat fleas suggested distinct functional changes characterizing these early Wolbachia host transitions. PPN Wolbachia had several putative horizontally transferred genes, including a lysine biosynthesis operon like that of the mitochondrial symbiont Midichloria , a spirochete-like thiamine synthesis operon shared only with w CfeT, an ATP/ADP carrier important in Rickettsia , and a eukaryote-like gene that may mediate plant systemic acquired resistance through the lysine-to-pipecolic acid system. The Discovery of group L-like variants from global rhizosphere databases suggests diverse PPN Wolbachia strains remain to be discovered. These findings support the hypothesis of plant-specialization as key to shaping early Wolbachia evolution and present new functional hypotheses, demonstrating promise for future genomics-based rhizosphere screens.