Deborah Loewenberg Ball

Effects of Teachers’ Mathematical Knowledge for Teaching on Student Achievement:

This study explored whether and how teachers’ mathematical knowledge for teaching contributes to gains in students’ mathematics achievement. The authors used a linear mixed-model methodology in which first and third graders’ mathematical achievement gains over a year were nested within teachers, who in turn were nested within schools. They found that teachers’ mathematical knowledge was significantly related to student achievement gains in both first and third grades after controlling for key student- and teacher-level covariates. This result, while consonant with findings from the educational production function literature, was obtained via a measure focusing on the specialized mathematical knowledge and skills used in teaching mathematics. This finding provides support for policy initiatives designed to improve students’ mathematics achievement by improving teachers’ mathematical knowledge.

The Mathematical Understandings That Prospective Teachers Bring to Teacher Education

This article focuses on the subject matter knowledge of preservice elementary and secondary mathematics teachers. In order to examine what teacher candidates understand about mathematics as they enter formal teacher education, results from questionnaires and interviews with 252 prospective teachers participating in a large study of teacher education are discussed. The results reveal the mathematical understandings that these elementary and secondary teacher candidates brought with them to teacher education from their precollege and college mathematics experiences, understandings that tended to be rule-bound and thin. Based on these data, the article challenges 3 common assumptions about learning to teach elementary or secondary mathematics: (1) that traditional school mathematics content is not difficult, (2) that precollege education provides teachers with much of what they need to know about mathematics, and (3) that majoring in mathematics ensures subject matter knowledge. These assumptions underlie current teacher education practices as well as proposals to reform the preparation of teachers.

Resources, Instruction, and Research

Many researchers who study the relations between school resources and student achievement have worked from a causal model, which typically is implicit. In this model, some resource or set of resources is the causal variable and student achievement is the outcome. In a few recent, more nuanced versions, resource effects depend on intervening influences on their use. We argue for a model in which the key causal agents are situated in instruction; achievement is their outcome. Conventional resources can enable or constrain the causal agents in instruction, thus moderating their impact on student achievement. Because these causal agents interact in ways that are unlikely to be sorted out by multivariate analysis of naturalistic data, experimental trials of distinctive instructional systems are more likely to offer solid evidence on instructional effects.

Developing Measures of Teachers’ Mathematics Knowledge for Teaching

In this article we discuss efforts to design and empirically test measures of teachers’ content knowledge for teaching elementary mathematics. We begin by reviewing the literature on teacher knowledge, noting how scholars have organized such knowledge. Next we describe survey items we wrote to represent knowledge for teaching mathematics and results from factor analysis and scaling work with these items. We found that teachers’ knowledge for teaching elementary mathematics was multidimensional and included knowledge of various mathematical topics (e.g., number and operations, algebra) and domains (e.g., knowledge of content, knowledge of students and content). The constructs indicated by factor analysis formed psychometrically acceptable scales.

The Work of Teaching and the Challenge for Teacher Education.

In this article, the authors argue for making practice the core of teachers’ professional preparation. They set the argument for teaching practice against the contemporary backdrop of a teacher education curriculum that is often centered not on the tasks and activities of teaching but on beliefs and knowledge, on orientations and commitments, and a policy environment preoccupied with recruitment and retention. The authors caution that the bias against detailed professional training that often pervades common views of teaching as idiosyncratic and independently creative impedes the improvement of teachers’ preparation for the work of teaching. They offer examples of what might be involved in teaching practice and conclude with a discussion of challenges of and resources for the enterprise.

Bridging Practices Intertwining Content and Pedagogy in Teaching and Learning to Teach

Subject matter and pedagogy have been peculiarly and persistently divided in the conceptualization and curriculum of teacher education and learning to teach. This fragmentation of practice leaves teachers on their own with the challenge of integrating subject matter knowledge and pedagogy in the contexts of their work. Yet, being able to do this is fundamental to engaging in the core tasks of teaching, and it is critical to being able to teach all students well. This article proposes three problems that would have to be solved to bridge this gap and to prepare teachers who not only know content but can make use of it to help all students learn. The first problem concerns identifying the content knowledge that matters for teaching, the second regards understanding how such knowledge needs to be held, and the third centers on what it takes to learn to use such knowledge in practice.

Relations Between Policy and Practice: A Commentary:

The research reported in the cases suggests that California’s new policy has affected instructional practice. Many teachers have tried to change their mathematics teaching, and some have made significant changes. But practice also has had a profound influence on the new policy. The teachers in the cases did not simply assimilate new texts and curriculum guides. They enacted new instructional policies in terms of their inherited beliefs, knowledge, and practices. Hence when teachers changed in response to the policy, they did so in terms of their pre-existing practice, knowledge, and beliefs. They reframed the policy in terms of what they already knew, believed, and did in classrooms. The result in many classrooms was a remarkable melange of old and new math teaching. This may be only the beginning of the story of the California math framework. It remains to be seen whether the reform will continue, and, if it does, whether the California system will be able to support this reform adequately.

Combining the Development of Practice and the Practice of Development in Teacher Education

This article describes and analyzes a program of work in elementary mathematics teacher education at the University of Michigan that has, for a decade, been a site for the development of approaches to preparing beginning K–8 mathematics teachers that are both aimed at practice and centered in content. Among the products of this work are video records, instructional tasks, and assessments, as well as structures for collective work on our courses. These materials and ways of working comprise a collection of resources for both the systematic improvement of the knowledge base for teacher education and the professional development of teacher educators. This practice‐based approach to the development of both our courses and their instructors has enabled us to improve professional instruction and to build knowledge that is useful beyond a particular course or the individuals who work in it. We begin by discussing 2 central problems of teacher education that our group has tried to address and then describe the types of materials and ways of working that we have developed. We analyze the features of the professional curriculum produced from this work, relate them to the problems, and consider issues about transferability to contexts beyond our own.

Reflections and Deflections of Policy: The Case of Carol Turner

This article presents a case of Carol, an accomplished second grade teacher who was disposed to teach in ways that seem consistent with the California Mathematics Curriculum Framework. Her approach-grounded in her conceptions of mathematics and her notions about how children learn mathematics—seemed, however, to have been virtually untouched by the policy initiative. Her practice, reflecting glimmers of the new ways, somehow also deflected them. Because the visions of mathematics and mathematical pedagogy represented in the Framework are multiple, Carol could be seen at once as complying with the Framework, or as subtly contradicting it. This raises questions both about the intentions of the Framework and about the nature of this reform. Would a state like California be happy if it could move all teachers to where Carol is? Alternatively, do the state policymakers want to change all teachers-those in the mainstream and on the fringe? The case highlights the complexity of the changes implied by California’s curriculum Framework and the difficulties inherent in communicating those changes in ways that can influence both pedestrian and accomplished practice.

Integrity in Teaching: Recognizing the Fusion of the Moral and Intellectual:

In this article we examine the relationship between teaching as a knowledge endeavor and teaching as a moral enterprise, using episodes from our own elementary school teaching as sites for our analysis. One episode concerns the teaching of social studies, the second the teaching of mathematics. We first describe the episodes themselves, highlighting the ways in which they shed light on issues of pedagogical content knowledge and reasoning. We then revisit each episode with a different lens: that of teaching as moral work. Our framework consists of two essential components: concerns for subject matter and for students. This analysis is meant to be neither a complete delineation of teaching as a moral enterprise nor an exhaustive analysis of pedagogical content knowledge. It is meant to show that, in teaching, concerns for the intellectual and the moral are ultimately inseparable.