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.]

Since it was signed into law, the Workforce Investment and Opportunity Act (WIOA) has been the subject of continuing scrutiny. One commonly shared concern is that the evaluation metrics of the Act (e.g., employment, salary, etc.) may incentivize programs to work with learners who will more readily meet expected outcomes (Pickard, 2016). Others suggest this concern is misplaced, pointing to the fact that the Act explicitly notes that the models created for State and program evaluation will be adjusted to recognize the priority given to serving students who face significant barriers (Wilson, 2015). These barriers include low levels of English and/or literacy, disabilities, limited work experience, lack of stable shelter, economic vulnerability and being an ex-offender. Learners who face these barriers are often identified as being part of "hard-to-serve" populations, both in policy documents and in commentary in the field. However, it is unclear what the term actually means. Because the term is not explained or elaborated on, "hard to serve" is open for interpretation. In this article, the author offers potential meanings for the term "hard to serve," and argues that recognizing there are no hard-to-serve students, only ill-served ones, will help clarify the political project required to remove the barriers that stand in these students' way.

The United States needs strong collaboration among adult educators and all social service agencies that support the linguistic, economic and civic integration of refugees and immigrants. Such collaboration can make possible holistic support required to create linkages between English language education and other non-educational support services. We provide examples of several interagency collaborative projects across the United States. Further, we argue that such collaboration is essential to mitigate the limitations of current adult basic education policy, which falls short of supporting linguistic integration of English language learners at the lowest proficiency levels, implicitly prioritizing workforce development programming best suited for higher-level learners. We layout policy recommendations for the local, state, and federal levels and map out benefits of working as partners in advocacy with agencies that support resettlement and integration of adult refugee and immigrant learners.

Each year in the United States, hundreds of thousands of people enroll in adult basic education (ABE) classes but leave before completing a level or accomplishing their goals. The persistence of this phenomenon may indicate that it is a feature of the system, rather than an unforeseen outcome. Research on other types of social service provision (e.g., welfare) suggests that seemingly inefficient systems are actually intentionally constructed to discipline the population in need of assistance. From this perspective, learners' experience of the churn within the ABE system may be just as important as their time in the classroom.

Instruments designed to measure teachers' knowledge for teaching mathematics have been widely used to evaluate the impact of professional development and to investigate the role of teachers' knowledge in teaching and student learning. These instruments assess a mixture of content knowledge and pedagogical content knowledge. However, little attention has been given to the content alignment between such instruments and curricular standards, particularly in regard to how content knowledge and pedagogical content knowledge items are distributed across mathematical topics. This article provides content maps for two widely used teacher assessment instruments in the USA relative to the widely adopted Common Core State Standards. This common reference enables comparisons of content alignment both between the instruments and between parallel forms within each instrument. The findings indicate that only a small number of items on both instruments are designed to capture teachers' pedagogical content knowledge and that the majority of these items are focused on curricular topics in the later grades rather than in the early grades. Furthermore, some forms designed for use as pre- and post-assessment of professional development or teacher education are not parallel in terms of curricular topics, so estimates of teachers' knowledge growth based on these forms may not mean what users assume. The implications of these findings for teacher educators and researchers who use teacher knowledge instruments are discussed.

Replication studies play a critical role in scientific accumulation of knowledge, yet replication studies in mathematics education are rare. In this study, the authors replicated Thanheiser's (Educational Studies in Mathematics 75:241-251, 2010) study of prospective elementary teachers' conceptions of multidigit number and examined the main claim that most elementary pre-service teachers think about digits incorrectly at least some of the time. Results indicated no statistically significant difference in the distribution of conceptions between the original and replication samples and, moreover, no statistically significant differences in the distribution of sub-conceptions among prospective teachers with the most common conception. These results suggest confidence is warranted both in the generality of the main claim and in the utility of the conceptions framework for describing prospective elementary teachers' conceptions of multidigit number. The report further contributes a framework for replication of mathematics education research adapted from the field of psychology.

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.

Contingent argument-based approaches to validity require a unique argument for each use, in contrast to more prescriptive approaches that identify the common kinds of validity evidence researchers should consider for every use. In this article, we evaluate our use of an approach that is both prescriptive "and" argument-based to develop a validity argument for a novel instrument. The elemental aspect concerns the items; the structural aspect concerns how the items function together; and the ecological aspect concerns how the resulting instrument is related to the context of use. Elemental, structural, and ecological aspects of validity are considered independently. We illustrate this approach by describing our experience constructing a validity argument for a novel, integrated knowledge and motivation instrument for use in teacher education. For each aspect, we illustrate the approach by discussing theory-based assumptions, articulating inferences based on these assumptions, and describing how we collected and assessed empirical evidence to support the inferences. We conclude with a retrospective analysis resulting in lessons learned for others who may consider a prescriptive, argument-based approach to validity.

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.

This study used units coordination as a theoretical lens to investigate how whole number and fraction reasoning may be related for preservice teachers at the conclusion of a math methods class. The study contributes quantitative evidence that units coordination provides a common foundation for both mathematical knowledge for teaching whole number addition and subtraction and for mathematical knowledge for teaching fraction multiplication and division. An analysis of PSTs' written responses to two different types of tasks (referent unit and appropriateness of proportional reasoning) revealed that some aspects of mathematical knowledge for teaching rational number and the multiplicative conceptual field are associated with units coordination (referent unit tasks), whereas others are not (appropriateness tasks). A logistic regression of PSTs' conceptions of multidigit number on their scores on these tasks provided empirical evidence that the most sophisticated conceptions of multidigit number (corresponding with having interiorized three levels of units) were associated with referent unit tasks but not associated with appropriateness of proportional reasoning tasks. Together, these findings support the claim that units coordination has promise for explaining the relationship between teachers' mathematical knowledge for teaching multidigit addition and subtraction and fraction multiplication and division. Implications for teacher education are discussed. [ABSTRACT FROM AUTHOR]

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.

This study (n = 1,044) used data from the Teacher Education and Development Study in Mathematics (TEDS-M) to examine the relationship between field experience focus (instruction- or exploration-focused), duration, and timing (early or not) and prospective elementary teachers' intertwined knowledge and beliefs about mathematics and mathematics learning. Early instruction-focused field experience (i.e., leading directly to classroom instruction) was positively related to the study outcomes in programs with such field experience of median or shorter duration. Moreover, the duration of instruction-focused field experience was positively related to study outcomes in programs without early instruction-focused field experience. By contrast, the duration of exploration-focused field experience (e.g., observation) was not related to the study outcomes. These findings suggest that field experience has important but largely overlooked relationships with prospective teachers' mathematical knowledge and beliefs. Implications for future research are discussed.

This chapter reviews how the impact of WIOA on adult education provision in the United States may be limited because of persistent structural problems within adult education policy and practice.

A very simple protocol for teaching Beer's Law and absorption spectrophotometry using a smart phone is described. Materials commonly found in high school chemistry laboratories or even around the house may be used. Data collection and analysis is quick and easy. Despite the simple nature of the experiment, excellent results can be achieved.

Mathematical knowledge for teaching (MKT) is often thought of as a transformed, mutually-influencing mixture of content and pedagogy. However when individuals' MKT does not integrate content and pedagogy, one type of knowledge can supersede the other, sometimes unconsciously. We exemplify this with Emma, a prospective elementary teacher, whose views on students' learning needs overshadowed her content knowledge. Specifically, Emma answered mathematical questions incorrectly, despite being aware of the correct answers, because she inadvertently gave more weight to pedagogical than to mathematical concerns. Implications for research and teacher education are discussed.

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.

This qualitative study examines critical pedagogy in Japanese adult basic education. The research focuses on what teachers and others think the current conditions are for education that deals with social justice. As part of this, the research looks at how critical pedagogy is conceptualized in this context. Participants in the study (literacy activists, teachers, students) expressed a concern that political shifts in Japan have made it more difficult to engage in social justice education. They also believe that structural changes in adult basic education (e.g., student demographics) have redefined what is possible. However, the study suggests that those involved in adult basic education are still working to create different types of educational spaces and that these spaces have particular political meanings. The suggestion is that spatial analysis may be a helpful way to investigate social justice work in different contexts.

Prior research suggests the interdependent nature of prospective teachers (PTs) beliefs about mathematics and beliefs about teaching and learning mathematics. Further, PTs' beliefs are said to be shaped by their prior experiences as mathematics learners. Using structural equation modeling, we assessed the relation between early childhood and elementary prospective teachers' (N = 213) prior school experiences with their Beliefs about Mathematics and Pedagogical Beliefs about teaching and learning mathematics as measured by the Integrating Mathematics and Pedagogy (IMAP) survey. The results suggest a significant direct relation between PTs' Beliefs about Mathematics and Pedagogical Beliefs. One main recommendation is that within the teacher preparation programs the classes should "speak" to each other as opposed to content being taught in one department and methods course in another department. In addition, results highlighted a non-significant indirect relation between PTs' Beliefs about Mathematics and Pedagogical Beliefs when mediated through their prior experiences. This signifies that regardless of prior experiences, PTs' Beliefs about Mathematics influence their Pedagogical Beliefs. This calls for teacher training programs to design learning experiences in a way that assists in shaping PTs beliefs irrespective of their experiences as elementary mathematics learners. [ABSTRACT FROM AUTHOR]

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