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The purpose of this study was to investigate how teachers understand one specific aspect of proportional reasoning. We were interested in understanding the extent to which practicing teachers were able to make sense of reasoning that involved the fixed number of variable-sized parts perspective. We used two items, drawn from a larger dataset, that encouraged teachers to reason about proportional situations using a variable parts perspective. We analyzed teachers' responses to these items using the knowledge in pieces perspective to identify fine-grained understandings teachers were applying to the situations. Our findings suggest that reasoning about variable-sized parts is not something all of these teachers did. Further, we found certain proportional structures that seemed to support teachers who were able to reason from this perspective. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the design of assessments of mathematics teachers' knowledge by emphasising the importance of identifying the purpose for the assessment, defining the specific construct to be measured, and considering the affordances of particular psychometric models on the development of assessments as well as how they are able to communicate learning or understanding. We add to the literature by providing illustrations of the interactions among these critical considerations in determining what inferences can be drawn from an assessment. We illustrate how the considerations shape assessments by discussing both existing and ongoing research projects. We feature discussion of two projects on which the authors of this paper are collaborating to demonstrate the affordances of attending to all three considerations in designing assessments of mathematics teachers' knowledge to provide readers with opportunities to see those considerations in use. [ABSTRACT FROM AUTHOR]

In this study, we investigated teachers' abilities correctly to identify situations that are appropriate for proportional reasoning and factors that might influence their ability using data on 148 teachers from the U.S. Learning Mathematics for Teaching assessment and other, similar items (with a smaller sample). Through a quantitative exploratory data analysis, we found middle school mathematics teachers are relatively proficient at correctly identifying situations that are proportional. However, we also found these same teachers were challenged by identifying situations as appropriate for proportional reasoning when in fact they were not, particularly in linear cases with a non-trivial constant term or inversely proportional situations. We also examined teachers' backgrounds and perceived expertise to determine whether there were correlations between their abilities to discern proportional situations and these attributes. We found that teachers are relatively proficient at self-assessing their expertise. Our findings imply that the middle school mathematics teachers in our study seemed to over-identify situations as proportional even when the situations were not proportional. [ABSTRACT FROM AUTHOR]

While there is considerable scholarship describing principles for effective professional development, there have been few attempts to examine these principles in practice. In this paper, we identify and examine the particular design features of a mathematics professional development experience provided for middle grades teachers over 14 weeks. The professional development was grounded in a set of mathematical tasks that each had one right answer, but multiple solution paths. The facilitator engaged participants in problem solving and encouraged participants to work collaboratively to explore different solution paths. Through analysis of this collaborative learning environment, we identified five design features for supporting teacher learning of important mathematics and pedagogy in a problem-solving setting. We discuss these design features in depth and illustrate them by presenting an elaborated example from the professional development. This study extends the existing guidance for the design of professional development by examining and operationalizing the relationships among research-based features of effective professional development and the enacted features of a particular design. [ABSTRACT FROM AUTHOR]

We report the development of a multiple-choice instrument that measures the mathematical knowledge needed for teaching arithmetic with fractions, decimals, and proportions. In particular, the instrument emphasizes the knowledge needed to reason about such arithmetic when numbers are embedded in problem situations. We administered our instrument to a convenience sample of 201 middle grades teachers and used the mixture Rasch model to uncover 2 distinct subgroups. Further analyses of the test data and follow-up interviews with 16 of the same teachers revealed that group membership provided 2 important pieces of information. First, teachers in Group 1 understood the role of multiplying or dividing by 1 in standard computation procedures much better than teachers in Group 2. Second, teachers in Group 1 identified appropriate referent units for numbers and appropriate parts-of-parts of quantities more consistently across situations than teachers in Group 2. Capturing such information extends recent advances in measuring mathematical knowledge for teaching. [ABSTRACT FROM AUTHOR]

The purpose of this research was to understand how one teacher reflected on different classroom situations and to understand whether the teacher's approach to these reflections changed over time. For the purposes of this study, we considered reflection as the teacher's act of interpreting her own practices and students 'thinking to make sense of student understanding and how teaching might relate to that understanding. We investigated a middle school mathematics teacher's reflection on her students while watching videotapes of her classroom and categorized the reflection as Assess, Interpret, Describe, Justify, and Extend. The results show a higher percentage of Extend instances in later interviews than in earlier ones indicating the teacher's increasing attention to her own teaching in how her students developed their understanding. In addition, her reflection became clearer and better integrated as defined by the Cohen and Ball's triangle of interactions. [ABSTRACT FROM AUTHOR]

This study considers how four groups of education graduate students worked in a distributed problem-based learning (PBL) environment using an asynchronous threaded discussion tool. The analyses consider how the students were able to interact with each other using the tool and how they worked together to define the problem and identify key issues to explore. The outcomes include some design considerations for tools to support these activities as well as insight about how students may work together to solve a problem in a distributed environment. [ABSTRACT FROM AUTHOR]

Variance and invariance are two powerful mathematical ideas to support geometrical and spatial thinking, yet there is limited research about teachers' knowledge of variance and invariance. In this paper, we examined how high school teachers deal with the task of looking for invariant properties in a dynamic geometry environment (DGE) setting. Specifically, we investigated if they even attend to invariant properties; what invariant properties they discern and discuss; and how DGE can support such discernment. Our analysis found that teachers tend to discern and discuss invariant properties mainly when they were probed to consider invariance. We also found four categories of invariant properties that seem to be important for a robust and rich understanding of geometric objects in the context of invariance and DGE. The use of DGE allowed teachers to see and interact with invariant properties, thus suggesting that accessing geometry dynamically may have structural affordances especially when exploring invariance. Teachers were able to enact different DGE movements to discern and discuss invariant properties, as well as to reason with and about them. We also saw that teachers' backgrounds and past experiences can play an important role in their descriptions of invariant properties. Possible future research directions and implications to teacher education are discussed. [ABSTRACT FROM AUTHOR]

Scholars and practitioners have called for personalized and widely accessible professional development (PD) for teachers. Yet, a long-standing tension between customizing support and increasing access to such support has hindered the scale-up of high-quality PD for individual teachers. This study addresses this challenge by developing a computerized program for middle school mathematics teachers that provides frequent opportunities for teachers to interact with and obtain personalized and real-time feedback from a virtual facilitator based on natural language processing. Based on the data collected from 1727 middle school students in an experiment in which the teachers of these students were randomly assigned to the program or the business-as-usual condition (i.e., the control group), we found that the program had a statistically significant impact on students' mathematics performance. These results demonstrate the potential of incorporating an automated, interactive feedback tool supported by artificial intelligence to create effective, scalable teacher PD. • An asynchronous, computerized teacher professional development (PD) program was designed for mathematics teachers. • Natural language processing was integrated into the PD system. • The program interacts with teachers and automatically provides personalized, timely feedback. • An experiment was conducted to estimate the impact of the program on students' mathematics performance. • The program improved students' mathematics performance. [ABSTRACT FROM AUTHOR]

This qualitative study considers middle grades mathematics teachers' reasoning about drawn representations of fractions and decimals. We analyzed teachers' strategies based on their response to multiple-choice tasks that required analysis of drawn representations. We found that teachers' flexibility with referent units played a significant role in understanding drawn representations with fractions and decimals. Teachers who could correctly identify or flexibly use the referent unit could better adapt their mathematical knowledge of fractions validating their choice, whereas teachers who did not attend to the referent unit demonstrated four problem-solving strategies for making sense of the tasks. These four approaches all proved to be limited in their generalizability, leading teachers to make incorrect assumptions about and choices on the tasks. [ABSTRACT FROM AUTHOR]

The article provides an insight into ways to solve open-ended problems by using multiple technologies. It states that technology is a powerful tool in assisting students in problem solving by allowing for multiple representations. It is particularly effective when students are encouraged to explore a single mathematics problem with multiple technologies. Moreover, using technology allows students to be creative in their problem-solving techniques, look at data in a variety of ways, and see the viability of their answers, all of which may lead to a better understanding of mathematics.

Researchers have recently used traditional item response theory (IRT) models to measure mathematical knowledge for teaching (MKT). Some studies (e.g., Hill, 2007; Izsák, Orrill, Cohen, & Brown, 2010), however, have reported subgroups when measuring middle-grades teachers' MKT, and such groups violate a key assumption of IRT models. This study investigated the utility of an alternative called the mixture Rasch model that allows for subgroups. The model was applied to middle-grades teachers' performance on pretests and posttests bracketing a 42-hour professional development course focused on drawn models for fraction arithmetic. Results from psychometric modeling and evidence from video-recorded interviews and professional development sessions suggested that there were 2 subgroups of middle-grades teachers, I better able to reason with 3-level unit structures and 1 constrained to 2-level unit structures. Some teachers, however, were easier to classify than others. [ABSTRACT FROM AUTHOR]