Mary Kay Stein

Building Student Capacity for Mathematical Thinking and Reasoning: An Analysis of Mathematical Tasks Used in Reform Classrooms

This article focuses on mathematical tasks as important vehicles for building student capacity for mathematical thinking and reasoning. A stratified random sample of 144 mathematical tasks used during reform-oriented instruction was analyzed in terms of (a) task features (number of solution strategies, number and kind of representations, and communication requirements) and (b) cognitive demands (e.g., memorization, the use of procedures with [and without] connections to concepts, the “doing of mathematics”). The findings suggest that teachers were selecting and setting up the kinds of tasks that reformers argue should lead to the development of students’ thinking capacities. During task implementation, the task features tended to remain consistent with how they were set up, but the cognitive demands of high-level tasks had a tendency to decline. The ways in which high-level tasks declined as well as factors associated with task changes from the set-up to implementation phase were explored.

Functions, Graphs, and Graphing: Tasks, Learning, and Teaching

This review of the introductory instructional substance of functions and graphs analyzes research on the interpretation and construction tasks associated with functions and some of their representations: algebraic, tabular, and graphical. The review also analyzes the nature of learning in terms of intuitions and misconceptions, and the plausible approaches to teaching through sequences, explanations, and examples. The topic is significant because of (a) the increased recognition of the organizing power of the concept of functions from middle school mathematics through more advanced topics in high school and college, and (b) the symbolic connections that represent potentials for increased understanding between graphical and algebraic worlds. This is a review of a specific part of the mathematics subject mailer and how it is learned and may be taught; this specificity reflects the issues raised by recent theoretical research concerning how specific context and content contribute to learning and meaning.

Instructional Tasks and the Development of Student Capacity to Think and Reason: An Analysis of the Relationship between Teaching and Learning in a Reform Mathematics Project

Abstract In the present study the relationship between teaching and learning was examined using a conceptual framework that links dimensions of instructional tasks with gains in student learning outcomes. The greatest student gains on a performance assessment consisting of tasks that require high levels of mathematical thinking and reasoning were related to the use of instructional tasks that engaged students in the “doing of mathematics” or the use of procedures with connections to meaning. In addition, student performance gains were greater for those sites whose tasks were both set up and implemented to encourage the use of multiple solution strategies, multiple representations, and explanations. Whereas, student performance gains were relatively small for those sites whose tasks tended to be both set up and implemented in a procedural manner and that required a single solution strategy, single representations, and little or no mathematical communication.

Leadership Content Knowledge.

Drawing inspiration from Shulman’s (1986) construct of pedagogical content knowledge, we propose that leadership content knowledge is a missing paradigm in the analysis of school and district leadership. After defining leadership content knowledge as that knowledge of academic subjects that is used by administrators when they function as instructional leaders, we present three cases of instructional leadership—situated at different school and district levels—and examine each for evidence of leadership content knowledge in use. Based on a cross-case analysis, we argue that as administrative levels increase and functions become broader, leadership content knowledge becomes less fine-grained, though always anchored in knowledge of the subject, how it is learned (by adults as well as students), and how it is taught. We go on to suggest that all administrators have solid mastery of at least one subject (and the learning and teaching of it) and that they develop expertise in other subjects by “postholing,” that is, conducting in-depth explorations of an important but bounded slice of the subject, how it is learned, and how it is taught. We conclude with an exploration of how content knowledge and leadership knowledge might be intertwined and suggestions for further research.

Subject-Matter Knowledge and Elementary Instruction: A Case from Functions and Graphing:

The purpose of the present investigation was to describe the relationship between teachers’ knowledge of mathematics and their instructional practice. An experienced fifth grade teacher was videotaped as he taught a lesson sequence on functions and graphing. In addition, a subject matter knowledge interview and card sort task were conducted with the teacher and a mathematics educator. The results suggest that the teacher’s knowledge of functions and graphing was missing several key mathematical ideas and that it was not organized in a manner to provide easily accessible, cross representational understanding of the domain. These limitations were found to relate to a narrowing of his instruction in three ways: the lack of provision of groundwork for future learning in this area, overemphasis of limited truths, and missed opportunities for fostering meaningful connections between key concepts and representations.

Architectures for Learning: A Comparative Analysis of Two Urban School Districts

This article explores the usefulness of communities of practice theory for understanding how districts can create organizational environments that foster teachers’ opportunities to learn the new ideas and practices required to carry out ambitious reforms. It draws on data from a longitudinal study of the implementation of ambitious mathematics curricula in two urban districts. By analyzing the contrasting ways that teachers in two schools in each district were linked to each other and reform efforts at the district level, the article shows how the district reform effort in one district led to significant opportunities for teacher learning and alignment with reform goals while efforts in the other district coordinated action but failed to spur meaningful opportunities for teacher learning. The article closes with implications for policy and practice.

Supporting Sustainability: Teachers' Advice Networks and Ambitious Instructional Reform.

Scaling up instructional improvement remains a central challenge for school systems. While existing research suggests that teachers’ social networks play a crucial role, we know little about what dimensions of teachers’ social networks matter for sustainability. Drawing from a longitudinal study of the scale-up of mathematics reform, we use qualitative social network analysis and qualitative comparative analysis (QCA) to investigate the relationship between teachers’ social networks and sustainability. Teachers’ social networks in the first 2 years of the initiative influenced their ability to sustain reform-related instructional approaches after supports for reform were withdrawn. Social networks with combinations of strong ties, high-depth interaction, and high expertise enabled teachers to adjust instruction to new conditions while maintaining the core pedagogical approach. This research contributes to our understanding of the dynamics of sustainability and to social network theory and research.

Algebra A Challenge at the Crossroads of Policy and Practice

The authors review what is known about early and universal algebra, including who is getting access to algebra and student outcomes associated with algebra course taking in general and specifically with universal algebra policies. The findings indicate that increasing numbers of students, some of whom are underprepared, are taking algebra earlier. At the same time, other students with requisite skills are not given access to algebra. Although studies using nationally representative data indicate strong positive outcomes for students who take algebra early, studies conducted only in contexts where all students are mandated to take algebra in eighth or ninth grade provide mixed evidence of positive outcomes, with increased achievement when policies include strong supports for struggling students. The authors conclude with a call for studies that examine the relationship among algebra policies, instruction, and student outcomes to understand the mechanisms by which policies can lead to success for all students.

Inquiry At the Crossroads of Policy and Learning: A Study of a District-Wide Literacy Initiative.

The literacy improvement efforts of New York City’s Community School District #2 serve as the locus of a study into the relationship between educational policy and practice. Based on 100 observations of classroom literacy instruction, a review of documentation related to the district’s Balanced Literacy Program, and interviews with teachers, staff developers, and district leaders the investigators found strong parallels between how the district children learn to read and district teachers learn to teach. These parallels are due in part to the ways in which District #2’s professional development system is anchored in the Balanced Literacy Program. They also stem from District #2 leaders’ beliefs in authentic and social forms of learning, beliefs that researchers found to have resonance with sociocultural theories of how individuals develop complex knowledge and skills. The result is a coherent system in which district policy regarding student learning is consistent with that of teacher learning.

Selecting and Supporting the Use of Mathematics Curricula at Scale

This article begins to unravel the question, “What curricular materials work best under what kinds of conditions?” The authors address this question from the point of view of teachers and their ability to implement mathematics curricula that place varying demands and provide varying levels of support for their learning. Specifically, the authors focus on how teacher capacity (their level of education, experience, and knowledge) and their use of curriculum influence instruction. The study sample is 48 teachers implementing two standards-based mathematics curricula—Everyday Mathematics and Investigations—in two school districts. The data include interviews and surveys with teachers, as well as observations of instruction, over a 2-year period. Findings indicate that teachers’ implementation of Investigations was considerably better than teachers’ implementation of Everyday Mathematics in terms of maintaining high levels of cognitive demand, attention to student thinking, and mathematical reasoning. These implementation measures were not correlated to measures of teacher capacity across school districts. However, implementation measures were significantly correlated with teachers’ lesson preparation that took into account the big mathematical ideas within curriculum. Further qualitative analysis indicated that the Investigations curriculum provided more support to teachers for locating and understanding the big mathematical ideas within lessons compared to Everyday Mathematics.