Ph.D. training across academic disciplines has faced significant challenges, with news and media outlets regularly highlighting student struggles, including frequent distrust and antagonism in relationships with faculty advisors. Given the centrality of the advisor-advisee relationship in doctoral education-most programs follow an apprenticeship model where students are mentored closely by a primary faculty advisor-the quality of this relationship may be particularly important. Yet almost no empirical research has systematically examined the long-term implications of advisor trust in doctoral education. In a prospective longitudinal study of 558 incoming Ph.D. students-primarily in STEM (Science, Technology, Engineering, and Mathematics) fields-at three US research universities, we examined the role of Ph.D. students' trust in their advisor early in graduate school on 17 outcomes measured repeatedly during students' first year, a sensitive period when a disproportionate number of students have historically exited. Results revealed a surprisingly broad and consistent effect of advisor trust, such that students who had greater trust in their advisor finished their first year more motivated, higher in well-being, and more academically successful than those with lower advisor trust. These effects were independent of student and advisor demographics, students' academic preparation, individual characteristics, and existing differences in outcomes measured at the start of graduate school, highlighting the potential causal implications of advisor trust for shaping a successful and healthy Ph.D. journey.

K-12 students need to become familiar with engineering because 21st-century careers integrate engineering practices across all science, technology, engineering, and mathematics (STEM) fields. While the Next Generation Science Standards (NGSS) emphasize learning real science and engineering practices, further work is needed to authenticate engineering for K-12 education. The NGSS are presented in a way that merges a single general practice with a core disciplinary idea and cross-cutting concept. Based on this framing and underlying epistemology, NGSS engineering practices are often implemented as overgeneralized, isolated, and largely context-neutral. Yet, in the STEM workplace, practices are rarely done in isolation from one another. Research is needed to better understand the integrated use of NGSS practices in STEM workplaces in order to make these practices more visible and accessible for use in K-12 classrooms. We conducted semi-structured interviews with 22 entry-level employees and managers in STEM-intensive workplaces. Using emergent and a priori coding, we identified how NGSS practices and contextual features are situated and connected in the STEM workplace. Our findings suggest that by adding rich contextual features, practices become more interconnected and reflective of authentic engineering in the STEM workplace. This process will enable educators to interpret K-12 engineering standards in an authentic way that supports deep engagement in learning engineering.

Considered a core component of children's foundational cognitive development, early mathematics experiences can support children's long-term academic success. Teachers and families alike share the common goal of wanting children to succeed developmentally, socially, and academically. Given the importance of early mathematics to academic success in all subjects, children need and deserve to build a robust knowledge of early math concepts in their earliest years. In this chapter, we consider the approach of the Young Mathematicians (YM) project at EDC, which for the past ten years, has partnered with families, teachers, and early childhood programs in richly diverse communities with large populations of students of color, linguistically minoritized students, and students living in poverty, to support math learning across home and school environments. We illustrate some of our fun early learning games that engage teachers and families alike and are freely available in multiple languages for anyone to use. We discuss how our close collaboration with families and teachers has informed our approach to equity and report on some of the positive results from our research. Finally, we reflect on ways we can all improve how we are partnering with families and teachers to create equitable and supportive learning communities.

In order to support physics students in their future careers, there is a need to understand the relationship between undergraduate education and professional practice in physics-related fields. This study investigated high-level goal driven mathematical problem-solving activities that are found within two disciplinary cultures: physical science research labs in academia and photonics workplaces in industry. We conducted semistructured interviews with 10 Ph.D. students and 22 engineers and technicians. Math use in professional workplaces was characterized through an adaptation of epistemic games framework, which revealed six common epistemic games in these workplaces: conceptual math modeling, analytical-numerical math modeling, design-oriented math modeling, fabrication, improving processes, and making meaning out of data games. The workplace-specific epistemic games capture the goals, starting and ending conditions, constraints and contextual features, moves, tools, and representations. The games involve a broad spectrum of math that ranges from arithmetic to computational modeling. The games reveal how goals and particular contextual features impact approaches to mathematical problem solving. The findings extend prior work on mathematical problem solving in physics to a new population of professional researchers, engineers, and technicians in their workplaces. The research may guide new approaches for developing problems and explicitly teaching problem solving in diverse physics contexts, which may additionally benefit undergraduate students' preparation for their future careers.

This study investigated how managers, entry-level employees, and hiring professionals in the optics and photonics industry socialize each other to enact the communication norms and expectations in their workplaces. A qualitative analysis of transcripts from interviews conducted with 33 employees at 15 companies produced five prevalent themes related to what optics and photonics employees consider competent communication (proactive questioning, efficient decision-making, familial-like humor, tactful translation, and fluent modality switching) and three socialization processes (presumed competence, informal mentoring, and structured training). These competencies and processes necessitate what we term cross-occupational communication: an interactive, iterative process involving communicative needs assessment, information exchange, and rhetorical/situational flexibility with groups distinct in background, training, and occupational role. It is difficult to create workplace-like experiences that truly capture the field-specific communication practices involved in organizational socialization within traditional classrooms; therefore, we argue for systematic and intentional communication in the disciplines instruction that considers cross-occupational communication needs in the workforce.

To prepare physics students for future careers, educators need to understand the skills and other factors necessary for entry and success on the job. Often these insights are based on CEOs, HR personnel, and managers, who provide broad perspectives regarding successful attributes of new hires. However, such insights are often more general and disconnected from specific jobs than insights gained from entry-level employees who recently transitioned from school to work. Using in-depth descriptive summaries and thematic analysis from interviews with six recently hired technicians and engineers in the field of optics, we explored factors that influenced their entry and success on the job. Six themes arose: documentation, computational skills, specialized learning, question asking, tinkering skills, and navigating cultural differences. We discuss the implications of these themes in both their value to employees as well as integration into the physics curriculum.

Problem-solving strategies learned by physics undergraduates should prepare them for real-world contexts as they transition from students to professionals. Yet, graduate students in physics-intensive research face problems that go beyond problem sets they experienced as undergraduates and are solved by different strategies than are typically learned in undergraduate coursework. This paper expands the notion of problem solving by characterizing the breadth of problems and problem-solving processes carried out by graduate students in physics-intensive research. We conducted semi-structured interviews with ten graduate students to determine the routine, difficult, and important problems they engage in and problem-solving strategies they found useful in their research. A qualitative typological analysis resulted in the creation of a three-dimensional framework: context, activity, and feature (that made the problem challenging). Problem contexts extended beyond theory and mathematics to include interactions with lab equipment, data, software, and people. Important and difficult contexts blended social and technical skills. Routine problem activities were typically well defined (e.g., troubleshooting), while difficult and important ones were more open ended and had multiple solution paths (e.g., evaluating options). In addition to broadening our understanding of problems faced by graduate students, our findings explore problem-solving strategies (e.g., breaking down problems, evaluating options, using test cases or approximations) and characteristics of successful problem solvers (e.g., initiative, persistence, and motivation). Our research provides evidence of the influence that problems students are exposed to have on the strategies they use and learn. Using this evidence, we have developed a preliminary framework for exploring problems from the solver's perspective. This framework will be examined and refined in future work. Understanding problems graduate students face and the strategies they use has implications for improving how we approach problem solving in undergraduate physics and physics education research.

Although there is widespread agreement on the importance of communication in physics-intensive careers, there is little prior research on how mathematics is integrated into communication. To examine how and what math is communicated in the workplace, we interviewed managers and recently hired employees from optics companies. Using a priori and emergent thematic coding, we found that symbolic math was purposefully hidden in many communication situations to avoid confusion. Alternatively, visual communication through blueprints, diagrams, and visuals of data was widely used across the optics workplace. Spreadsheets were a universal tool for exchanging mathematics through data, formulas, graphs, and calculations; however, different people relied on the spreadsheets for different purposes (e.g., executing calculations or programming formulas). Our data illuminates the value of specific strategies for communicating math and how math is communicated between employees, managers, and clients. These findings suggest the importance of teaching physics majors how to explain mathematical ideas for a variety of audiences and purposes.

It is important to develop models about how mathematics is used in professional physics settings. Existing models of math use focus on mathematical modeling for problem solving. However, workplace problems often include design problems, troubleshooting, and more. To study workplace mathematics, we conducted hour-long, semi-structured interviews with employees at photonics and optics companies in Rochester, NY. We applied an emergent coding process to classify instances of math in the workplace, and present two models of mathematics use within workplace tasks. We describe a four-phase engineer task consisting of defining the problem, designing a product, testing the product, and communicating results. A common technician task replaces the design phase with manufacturing the product. Workplace math is embedded in these phases through various representations such as simulations, schematics, and machining codes. Educators should consider using diverse problem types since they require additional mathematical representations and techniques to be brought to the forefront.

As physics departments increasingly emphasize computational training within the physics curriculum, there is a need for educators to have guiding principles for deciding how and when to use computational approaches over analytical math and vice versa. We investigated the use of analytical and computational mathematics in professional practice by conducting ten semi-structured interviews with PhD students in the physical sciences. The interviews revealed context-rich situations where computational and analytical math were valued and used. Through an emergent and thematic coding process, key contextual features were distilled. Although analytical math was valued as a calculational tool (e.g., manipulating equations), the most prevalent use of analytical math was to develop a preliminary understanding of a problem, which included modeling systems through equations, developing simplified toy models, understanding background concepts, and understanding how varying parameters affected system behavior. Computational tools had a complementary role of data analysis, complex numerical simulations, and visualization.