Search

Teachers' beliefs can have powerful consequences on instructional decisions and student learning. However, little research focuses on how teachers' beliefs about the role of race and gender in mathematics teaching and learning influence educational equity within classrooms. This gap is partly due to the lack of studies focused on variation within classrooms, which in turn is hampered by the lack of instruments designed to measure mathematics-specific equity beliefs. In this study of 313 preservice and practicing elementary teachers, we report evidence of construct validity for the Attributions of Mathematical Excellence Scale. Factor analyses provide support for a four-factor structure, including genetic, social, personal, and educational attributions. The findings suggest that the same system of attribution beliefs underlies both racial and gender prejudice among elementary mathematics teachers. The Attributions of Mathematical Excellence Scale has the potential to provide a useful outcome measure for equity-focused interventions in teacher education and professional development. [Note: The correct volume is 8, and the correct issue number is 1.]

We report a novel survey that narrows the gap between information about teachers' knowledge of fraction arithmetic provided, on the one hand, by measures practical to administer at scale and, on the other, by close analysis of moment-to-moment cognition. In particular, the survey measured components that would support reasoning directly with measured quantities, not by executing computational algorithms, to solve problems. These components--each of which was grounded in past research--were attention to referent units, partitioning and iterating, appropriateness, and reversibility. A second part of the survey asked about teachers' professional preparation and history. We administered the survey to a national sample of in-service middle-grades mathematics teachers in the United States and received responses from 990 of those teachers. We analyzed responses to items in the first part of the survey using the log-linear diagnostic classification model to estimate each teacher's profile of strengths and weaknesses with respect to the four components of reasoning. We report on the diversity of profiles that we found and on relationships between those profiles and various aspects of teachers' professional preparation and history. Our results provide insight into teachers' knowledge resources for enacting standards-based instruction in fraction arithmetic and an example of new possibilities for mathematics education research afforded by recent advances in psychometric modeling.

This study explores the dimensionality of mathematical knowledge needed by elementary school teachers. Specifically, we focus on the construct of Mathematical Knowledge for Teaching to investigate whether common mathematical content knowledge (the generic mathematical knowledge that is held by an educated adult), specialized content knowledge (the content knowledge that is unique to teaching), and pedagogical content knowledge (knowledge of the content, students, and teaching) can be distinguished empirically. Findings from a secondary analysis of data collected from upper elementary teachers in the Measures of Effective Teaching project (N = 397) suggest the most parsimonious explanation of Mathematical Knowledge for Teaching construct is that it is unidimensional. We argue, based on the findings and limitations we identified, that researchers should reconsider both the structure of the components that make up the content knowledge for teaching and the effects of item design on the dimensionality of the construct.

Mathematics teacher education aims to improve teachers' use of mathematical knowledge to support teaching and learning, an aspect of pedagogical content knowledge (PCK). In this study, we interviewed teachers to understand how they used mathematics to make sense of student solutions to proportional reasoning problems. The larger purpose was to find accurate ways of categorizing teachers' ability to do this vital aspect of teaching and thereby to inform assessment, teacher education, and professional development. We conjectured that teachers' PCK for proportional reasoning could be reliably described in terms of attention to quantitative meanings in story problem contexts and in terms of understanding naïve forms of proportional reasoning. Instead, our findings reveal that individual teachers used a variety of means to make sense of (1) cognitively similar student solutions to different tasks and (2) mathematically related steps of a student solution within a single task. These findings illustrate the complexity of PCK for the topic of proportional reasoning and suggest the limits of what can be inferred about teacher knowledge from teachers' evaluations of student solutions. We discuss implications for teacher education and assessment.

The Mathematics Scan (M-Scan), a content-specific observational measure, was utilized to examine the extent to which "standards-based mathematics teaching practices" were present in three focal lessons. While previous studies have provided evidence of validity of the inferences drawn from M-Scan data, no prior work has investigated the affordances and limitations of the M-Scan in capturing standards-based mathematics teaching. We organize the affordances and limitations into three categories: the operationalization of the M-Scan, the organization of the M-Scan, and the M-Scan within the larger ecology of instruction. Our analysis indicates the M-Scan differentiates among lessons in their use of "standards-based mathematics teaching practices" by operationalizing the M-Scan dimensions at the lesson level, sometimes at the expense of capturing the peaks and valleys within a single lesson. Simultaneously, the analysis revealed how the application of the rubrics may be impacted by lesson transcripts. We discuss the theoretical organization of the M-Scan and its implications for researchers and practitioners applying the rubrics. Finally, we point to the affordances and limitations of the M-Scan within the larger ecology of instruction by considering curricular issues and two dimensions of instruction not highlighted by the M-Scan.

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.

This paper focuses on the design of tasks that situate teachers in the work of mathematics teaching. We examine a simple, yet fundamental feature of pedagogical content knowledge tasks: the extent to which pedagogical warrants are necessitated by the pedagogical context. We argue that this feature has implications for the design of tasks, as well as implications for understanding how pedagogical content knowledge is used to reason about these tasks. To develop this argument, we use representative cases of interviews of teachers and mathematicians to analyze how the pedagogical context of the tasks shaped opportunities for eliciting pedagogical content knowledge.

Past studies have documented students' and teachers' persistent difficulties in determining whether 2 quantities covary in a direct proportion, especially when presented missing-value word problems. In the current study, we combine a mathematical analysis with a psychological perspective to offer a new explanation for such difficulties. The mathematical analysis highlights numbers of equal-sized groups and places reasoning about proportional relationships in the context of reasoning about multiplicative relationships more generally. The psychological perspective is rooted in diSessa's (diSessa, 1993, 2006; diSessa, Sherin, & Levin, 2016) knowledge-in-pieces epistemology and highlights diverse, fine-grained knowledge resources that can support inferring and reasoning with equal-sized groups. We illustrate how the combination of mathematical analysis and psychological perspective may be applied to data using empirical examples drawn from interviews during which preservice middle-grades teachers reasoned with varying degrees of success about relationships presented in word problems that were and were not proportional.

Past studies have suggested that in light of recent curriculum standards, many US teachers make limited use of drawn models in their mathematics instruction. To gain insight into this phenomenon, we investigated relationships between US teachers' opportunities to learn about, knowledge of, motivation for, and instructional use of drawn models for representing multiplication and division of fractions. A national sample of 990 practicing middle-grade teachers was administered a three-part survey that contained a knowledge assessment; a professional history and teaching practices questionnaire that included questions about opportunities to learn to use drawn models; and a motivation questionnaire that measured teachers' value, anxiety, and self-concept of ability for using such models in instruction. In regression models without motivation, opportunity to learn significantly predicted the teachers' knowledge, frequency of use, and purpose for use of drawn models. In structural equation models that included motivation, knowledge and motivation substantially accounted for relationships between the teachers' opportunity to learn and their self-reported use of drawn models in instruction. These findings are consistent with the general hypothesis that teacher' opportunities to learn teaching practices indirectly affect their instructional practices. Teachers' knowledge and motivation also play a central role.

For many students, making connections between mathematical ideas and the real world is one of the most intriguing and rewarding aspects of the study of mathematics. In the Common Core State Standards for Mathematics (CCSSI 2010), mathematical modeling is highlighted as a mathematical practice standard for all grades. To engage in mathematical modeling, beginning algebra students must learn to use their understanding of arithmetic operations to make mathematical sense of problem situations and to relate this sense making to functions represented by equations, tables, and graphs. The word problems commonly used in beginning algebra courses give opportunities to practice mathematical modeling. Further, the ability to reason with quantities as well as numbers is an important capacity for students to develop. Two kinds of quantitative reasoning have a special relevance for beginning algebra students. The "correspondence" perspective deals with the question, How is one quantity related to another? A correspondence understanding of speed might be expressed as the rule that relates each value for time with a unique value for distance, such as the equation y = 25x, where x represents time and y represents distance. By contrast, the key question for "covariation" reasoning is, How does one quantity change as another quantity changes? A covariation understanding of speed would focus on how distance and time change together--that is, the distance covered increases by 25 meters as the elapsed time increases by 1 second. Both kinds of reasoning are important goals for algebra students. Correspondence is a fundamental piece of mature reasoning about functions, and covariation is critical for developing the rate-of-change concept. Presented in this article are two sessions from Ms. Holmes's classroom (the teacher's name is a pseudonym) in which seventh graders intuitively used covariation to begin to make sense of word problems. The passages show how students' covariation reasoning might surface in the classroom and illustrate some of the teaching strategies that Ms. Holmes used to support her students' reasoning. The sessions also provide a foundation for the discussion of classroom strategies, which summarizes research-based strategies for supporting students' use of covariation reasoning to build robust mathematical models.

In this qualitative research study, we sought to understand teachers' conceptions of integrated mathematics. The participants were teachers in the first year of implementation of a state-mandated, high school integrated mathematics curriculum. The primary data sources for this study included focus group and individual interviews. Through our analysis, we found that the teachers had varied conceptions of what the term integrated meant in reference to mathematics curricula. These varied conceptions led to the development of the Conceptions of Integrated Mathematics Curricula Framework describing the different conceptions of integrated mathematics held by the teachers. The four conceptions--integration by strands, integration by topics, interdisciplinary integration, and contextual integration--refer to the different ideas teachers connect as well as the time frame over which these connections are emphasized. The results indicate that even when teachers use the same integrated mathematics curriculum, they may have varying conceptions of which ideas they are supposed to connect and how these connections can be emphasized. These varied conceptions of integration among teachers may lead students to experience the same adopted curriculum in very different ways.

Researchers have recently used traditional item response theory (IRT) models to measure mathematical knowledge for teaching (MKT). Some studies (e.g., Hill, 2007; Izsak, 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, 1 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. (Contains 12 figures, 4 footnotes and 2 tables