Erik Jacobson

We report a multidimensional test that examines middle grades teachers’ understanding of fraction arithmetic, especially multiplication and division. The test is based on four attributes identified through an analysis of the extensive mathematics education research literature on teachers’ and students’ reasoning in this content area. We administered the test to a national sample of 990 in‐service middle grades teachers and analyzed the item responses using the log‐linear cognitive diagnosis model. We report the diagnostic quality of the test at the item level, mastery classifications for teachers, and attribute relationships. Our results demonstrate that, when a test is grounded in research on cognition and is designed to be multidimensional from the onset, it is possible to use diagnostic classification models to detect distinct patterns of attribute mastery.

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.

Proportional reasoning comprises a network of understandings and relationships, and it represents a milestone in students' cognitive development (Lamon, 2007; Vergnaud, 1983). Furthermore, proportional reasoning is foundational to students’ later development of concepts related to functions, graphing, algebraic equations, and measurement (Karplus, Pulos, & Stage, 1983; Lobato & Ellis, 2010; Thompson & Saldanha, 2003). Because proportional reasoning is complex, helping students develop the associated big ideas and essential understandings is not easy. This pedagogical task involves deepening one’s own understanding as a teacher and being sensitive to the types of reasoning that are most accessible as entry points for students while pushing them to develop more sophisticated forms of reasoning.However, research on teachers’ knowledge in this domain has primarily bootstrapped models of students’ proportional reasoning to assess and make sense of teachers’ understanding, consequently reporting similar sources of conceptual difficulty for teachers as for students (eg, additive reasoning)(Ben-Chaim, Keret, & Ilany, 2007; Cramer, Post, & Currier, 1993; Hull, 2000). Consequently, several important questions emerge:(a) Do middle school teachers have conceptual resources related to proportionality that extend beyond what students typically have and which can be useful to the task of helping students develop proportional reasoning; and (b) Is teachers’ mathematical knowledge for teaching proportional reasoning marked by any conceptual challenges that differ from the typical challenges and misconceptions identified for …

Teacher affect heavily influences instruction and learning (Cross 2009; Pajares 1992; Philipp 2007; Robertson-Kraft and Duckworth 2014). Teacher affect, which includes partially cognitive traits such as attitudes and beliefs as well as noncognitive traits such as emotion, motivation, and grit, is often defined in opposition to purely cognitive traits such as IQ or mathematical knowledge. As a consequence, teacher affect has often been studied in isolation from teacher cognition (Philipp 2007; Thompson 1992). By contrast, some researchers collapse cognitive and partially cognitive categories, grouping mathematics teacher knowledge and beliefs together as beliefs (eg, Leatham 2006) or as knowledge (eg, Beswick et al. 2012). Mathematical knowledge for teaching (MKT), one of the most widely used frameworks for teacher knowledge, is defined to comprise ‘‘skills, habits, sensibilities as well as knowledge’’(Ball et al …

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 …

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 …

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 authors 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. Findings suggest that field experience has important but largely overlooked relationships with prospective teachers' mathematical knowledge and beliefs. Implications for future research are discussed.

Mathematics education researchers in several recent projects have used psychometric models to develop measures of newly conceptualized domains of content knowledge for teaching. In this study, we conducted a retrospective analysis across four projects to investigate the interactive process of domain conceptualization and measure design. Our analysis uncovered three productive tensions that researchers have experienced in their work, and we used an argument-based framework for validity to describe how the interaction between measure design and construct development allowed the resolution of each of these tensions. Our results suggest that developing measures of newly conceptualized domains is a form of inquiry, and we propose that future projects engaged in similar work might benefit from an explicit focus on the assumptions and inferences of a validity argument.

The purpose of this dissertation was to investigate the role of background, experience, and interactions with colleagues and students in the development of mathematical proficiency for teaching, operationalized as teachers’ content knowledge for teaching and beliefs about learning and mathematics. I conducted three studies in different contexts. The purpose of the first study of Texas K–12 mathematics teachers in their first 5 years of teaching was to describe how mathematical proficiency for teaching multiplicative reasoning varied across preparation, school contexts, and a wide range of grade levels.Surprising findings from the first study led to two follow-up studies:(1) content knowledge for teaching was not positively related to length of teaching experience and (2) the length of student teaching had no significant relationship to mathematical proficiency for teaching. The second study used a longitudinal design to study change in Grades 6–8 teachers’ mathematical proficiency for teaching multiplicative reasoning topics. I found that teachers’ content knowledge for teaching increased over the semester, especially for teachers with less mathematical preparation, but that their self-efficacy beliefs decreased. In the third study, I

Table representations of functions allow students to compare rows as well as values in the same row.

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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 …

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 …