Teachers' knowledge of the subject matter is considered an important component of their expertise in teaching mathematics. Yet how teachers' understanding of one content area is related to other content areas has not been investigated in depth. We explored this question by investigating teachers' knowledge of two theoretically related areas: (1) fractions and (2) ratios and proportional relationships. We also investigated the extent to which teachers' educational backgrounds are related to their understanding of these concepts. Based on the results obtained from structural equation modeling and path analysis, we found that teachers' knowledge of these two concepts is highly interdependent, forming a single construct. Furthermore, holding a credential in teaching mathematics, the route teachers took to enter teaching, and their undergraduate majors were associated with their knowledge of these concepts. This study illustrates the importance of attending to the theoretical relationships among different content areas when assessing teachers' subject matter knowledge and provides initial evidence that teachers' subject matter knowledge may be unidimensional for theoretically related domains.

The factors to which teachers attribute students’ successes and failures have important consequences for teachers’ instructional decisions as well as their expectations of students. Yet few, if any, experimental studies have been conducted to investigate the role that students’ race and gender play in how teachers make sense of students’ academic performance. In this exploratory experimental study, conducted with 413 mathematics teachers across the United States, we investigated the extent to which teachers attributed students’ performance disparities in mathematics to internal and external factors based on the students’ race and gender and explored the extent to which teachers’ beliefs and dispositions moderated their attributions. To do so, teachers were first given a collection of the student solutions with no gender or racial performance disparities. They were then randomly assigned to one of four conditions in which they were told that on this particular assessment: boys outperformed girls, girls outperformed boys, Black and Hispanic students outperformed White and Asian students, or that White and Asian students outperformed Black and Hispanic students. They were asked to report the reasons behind the disparities in math performance of these groups. We found that teachers attributed gender disparities in student performance to innate math ability, effort, and external social factors, whereas they attributed racial disparities in student performance to biological influences on intelligence, effort, external social factors or the assessment context. Teachers’ self-reported personal experience with racial discrimination moderated race differences in teacher attributions. © 2024 The Authors

While research has shown that students benefit from student-centered pedagogies, few studies have considered the benefits of this pedagogical approach for educators as they learn through teaching. In response to this need, we analyzed interviews, lesson plans, and video observations from five teachers in elementary schools across the United States who varyingly engaged student-centered and teacher-centered pedagogies. Our analyses revealed that the participating teachers developed a wide breadth of teacher knowledge regardless of their pedagogical approach. However, the teachers who employed student-centered teaching reported more pedagogical content knowledge gains for themselves than the teachers who used direct teaching.

Pedagogical content knowledge (PCK) has been widely recognised as an important aspect of the expertise for teaching. However, the extent to which teachers' own teaching practice can be a learning resource for them to develop PCK has not been systematically explored. This empirical study aimed to explore the unique contribution of the work of teaching in teachers' PCK growth by concurrently considering other external professional learning opportunities teachers may have on the job. Using longitudinal data from 207 elementary and middle school teachers in the United States, we found that teachers increased their PCK through teaching on their own, albeit at different rates. Our findings were robust when other external learning opportunities teachers had were taken into account. Our findings underscored the importance of teachers' robust knowledge of school mathematics in the development of their PCK through teaching.

Background: What and how teachers learn through teaching without external guidance has long been of interest to researchers. Yet limited research has been conducted to investigate how learning through teaching occurs. The microgenetic approach (Siegler and Crowley, American Psychologist 46:606–620, 1991) has been useful in identifying the process of student learning. Using this approach, we investigated the development of teacher knowledge through teaching as well as which factors hinder or promote such development. Results: Our findings suggest that teachers developed various components of teacher knowledge through teaching without external professional guidance. Further, we found that the extent to which teachers gained content-free or content-specific knowledge through teaching depended on their robust understanding of the concept being taught (i.e., content knowledge), the cognitive demand of the tasks used in teaching, and the lesson structure chosen (i.e., student centered vs. teacher centered). Conclusions: In this study, we explored teacher learning through teaching and identified the sources leading to such learning. Our findings underscore the importance of teachers’ robust understanding of the content being taught, the tasks used in teaching, and a lesson structure that promotes teachers’ learning through teaching on their own. © The Author(s) 2024.

Teachers' mathematical knowledge has important consequences for the quality of the learning environment they create for their students to learn mathematics. Yet relatively little is known about how teachers reason proportionally, despite the fact that proportional reasoning is foundational for several mathematics concepts and that ratios and proportional relationships constitute a major component of the middle school mathematics curriculum. In this study, we investigated how teachers reasoned proportionally on a nonroutine ratio task and the extent to which their proportional reasoning was able to predict their overall understanding of the relevant concepts: ratios and proportional relationships. Using data collected from 238 US mathematics teachers, we found that teachers' proportional reasoning could be grouped into four categories: incorrect, additive, relative, and proportional reasoning. Our results also indicated that teachers' overall knowledge of ratios and proportional relationships aligned with the way they reasoned proportionally, meaning that teachers who used incorrect reasoning on a separate task received the lowest scores on average on the ratios and proportional relationships measure, whereas those who reasoned proportionally had the highest mean scores on average. Implications of the study include the need to shift attention to the way teachers reason in relation to the two elements of proportional reasoning (covariance and invariance) to capture the nuances in their understanding of ratios and proportional relationships.

Teaching a particular subject provides rich opportunities for teachers to develop subject-specific knowledge and skills, especially for those who are in the early years of their career. Yet supporting evidence is scarce regarding the extent to which knowledge and skills teachers could gain from their teaching experience. This study aims to address this gap by collecting data from a from a national sample of 207 novice mathematics teachers for three years in a row to explore the development of two elements of teachers' pedagogical content knowledge (PCK) of mathematics: (1) knowledge of students' mathematical thinking and (2) knowledge of mathematics teaching. By using linear growth modeling to analyze the data derived from teachers' analyses of videos clips of mathematics classes, we found that teachers increased both elements of PCK, albeit at different rates. Further, the growth of these two important elements of PCK were associated with different teacher-related factors. Having a robust knowledge of mathematics played a key role in teachers' learning of students' mathematical thinking, whereas having a credential in mathematics teaching played a role in the development of teachers' knowledge of mathematics teaching. (c) 2022 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

This study investigates the applicability of topic modeling to analyze educational data. Topic modeling is useful because it reveals the latent topic structure underlying a collection of texts. Because metadata provides useful information about the topics, this study explores a way of including metadata as a covariate predicting topics and outcomes by topic in a topic model using a two-step approach. In Study 1, we use structural topic model (STM) and regression model because STM estimates the topic structure and how covariate is related to the topics. In Study 2, supervised Dirichlet allocation (sLDA) model is used to investigate the relationship between topics and the outcome variable: we incorporate sLDA with ANOVA. We demonstrate that the inclusion of multiple metadata improved the interpretability of the topic modeling techniques' results by examining the relationship among the examinees' written answers, problem-solving strategies, and scores using the empirical data of 246 middle school mathematics teachers' written responses to an item.

This article explores three attributes of teachers' understanding of fraction magnitude: the accuracy and reasonableness of teachers' estimations in response to fraction arithmetic tasks as well as the alignment of the estimation strategies they used with the concept of fraction magnitude. The data were collected from a national sample of mathematics teachers in grades 3-7 in which fraction concepts were taught (N = 603). The results indicated the teachers' estimations were only partially accurate and reasonable, particularly when fraction division was involved. Furthermore, teachers' credentials and the grade level at which they taught mathematics were significantly related to teachers' understanding of fraction magnitude.

Attending to the whole unit that a number refers to in a mathematical problem situation and showing flexibility in coordinating different units are foundational for mathematical understanding. In this study, we explored teachers' attention to and flexibility with referent units in situations involving fractions and fraction multiplication. Using data collected across the USA from 246 mathematics teachers in Grades 3-7 where fractions are taught, we found that teachers' attention to and flexibility with referent units were related to each other as well as to teachers' overall knowledge of fractions.