Project Abstract

This project seeks to understand middle grade students’ transition from arithmetic into algebraic reasoning, and to develop and evaluate proper educational approaches to improve the learning and teaching of increasingly complex mathematics. Acknowledging the complexity of this area of study, we have designed a comprehensive, systemic research and development program to address three inter-related areas of study, or tiers: Student learning and development; teacher beliefs, knowledge and practice; and teacher professional development (TPD). Our approach emphasizes the parallel structures and processes among these tiers, as distinct but inseparable aspects of a unified system. Our specific research activities reflect this multi-tiered and dynamic framework, as an attempt to move beyond piecemeal, disconnected insights to reach a deeper appreciation of the learning and teaching processes we set out to study, and to develop a coherent program of instruction and TPD.

We argue that improvement of algebra education must be grounded in a theory of how students develop algebraic reasoning and acquire domain knowledge and skills, and the beliefs and existing practices of teachers. The theory is intended to inform curriculum design, assessment, teaching practices, and TPD efforts.

Our methodological approach is both naturalistic and experimental, and draws on situated observations, case development and teaching experiments, as well as quasi-experimental design.

Within the vast domain of school algebra, we focus on the core concepts of equality and variable, and the competencies of generalizing and representing patterns and functions, complex problem solving, and fluency with formal representations. Identification of these foci guides assessment design, classroom observations, and the TPD program. Our student assessment process focuses on four primary activities: (1) defining and specifying the specific areas within the domain of algebra to be assessed, (2) identifying items from traditional and reform curricula, published studies as well as generation of new items to address these focus areas, (3) providing written assessments to large samples of 6th, 7th, and 8th grade students in urban Denver, CO, rural Brighton, CO, and Madison, WI, (4) interviewing individual students in one-on-one settings about their approaches to a subset of the items, and (5) developing embedded assessment activities that can be used by teachers in classrooms.

Our assessment activities include cross-sectional as well as longitudinal designs across the middle school grades that are designed to provide diagnostic information for the development of a class of models for the transition to abstract, symbolic algebraic thinking. Guided by explicit developmental models, assessment tasks are constructed to elicit variations in student performance indicating different levels of competence at different developmental stages. The tasks are grounded in familiar classroom activities students are likely to encounter in their mathematics courses. Students are asked to identify, interpret, generate, and manipulate a variety of mathematical problem representations: verbal, tabular, graphical, and symbolic. Systematically varying prominent aspects of the tasks (such as the complexity of mathematical relations, the level of abstraction provided and asked for, and the initial representational format) allows us to examine specific developmental hypotheses. Analysis of student performance and reasoning aims to describe typical performance and individual differences within each developmental stage, as well as characterize transitions between stages. The analysis focuses on evidence of proficiency, quality of explanations, strategy use, and fluency with and among different representations.

The student tier is designed to support the construction of a developmental model of students’ evolving algebraic reasoning and skill acquisition within each of the focal areas, paying close attention to the transition from intuitive, verbally-grounded reasoning and concrete problem-solving strategies to comprehension and mastery of the formal, abstract methods of algebraic reasoning. This will be used to inform the next generation of Algebra Cognitive Tutor for the middle grades (see www.carnegielearning.com), which provides a vehicle for scaling up to large numbers of classrooms.

In the teaching tier, we examine the effects of both traditional and reform (Connected Mathematics and Mathematics in Context) curricula on students’ algebraic development (as measured by recurring longitudinal assessment) and on teachers’ beliefs, instructional practices and TPD. We also explore teachers’ affective and cognitive views regarding student knowledge and learning, as well as how teachers perceive their own practices. We will first set out to better understand issues related to the teaching of algebraic concepts in middle school classrooms, paying particular attention to the struggles middle school teachers (often trained in elementary education) face implementing advanced mathematics, and how they address new curricular and professional practices. Several research questions are of interest. How do teachers define and understand algebra? How do teachers understand and engage in new, reform-based curricular programs where algebraic concepts are central? How do teachers’ conceptions of algebra (and reform-based algebra curricula) impact their teaching practices? What sorts of pedagogical challenges await teachers in these reform classrooms and how do they cope with these challenges? To pursue these questions, we will develop case studies of 12 teachers in three school districts that include inner city and rural communities. In addition to using belief survey instruments (administered twice during the year), we will collect data through classroom observations, perform structured interviews and pre- and post-lesson conferences, and collect teachers’ written reflections. This rich set of data will be used to describe teachers and their practices along several dimensions (e.g., algebra content knowledge, beliefs about the teaching of algebra, and teachers' tolerance for discomfort) and thereby help us to better understand the landscape of middle school algebra instruction during this era of reform.

In the teacher professional development tier, we will design, implement and evaluate a "proof-of-concept" TPD program, building on our cumulative insights in the other tiers about the domain of school algebra, student reasoning and development, and teachers’ knowledge and beliefs about students’ understandings of algebra. The focus will be on those aspects identified as important for supporting students’ transition from arithmetic to algebraic reasoning. The TPD program will also focus on ways in which these beliefs play out in teachers’ practices. Issues raised by the teaching tier about teacher discomfort with mathematical content and teaching techniques, as well as links between instructional practices and state or district standards for teaching mathematics will also be addressed.

The aim of our TPD research is to evaluate and perfect a scalable model of TPD designed to impact teacher practice on a broad scale, moving the results of this research into many middle school classrooms. Our approach, which will be implemented in years 4 and 5 of our project, will be based on recent theories of teacher learning that articulate the importance of engaging teachers’ prior knowledge and beliefs in order to challenge them or build upon them with progressive, new ideas about teaching and learning. These theories also highlight the advantages of situating teachers’ learning experiences in classroom contexts and in social interaction with colleagues. Our model of TPD will thus integrate activities focused on enactments of new ideas in the classroom, structured reflection on the problems and successes experienced during enactment, and professional discourse and collaboration. On-line video cases and discussion groups will permit large-scale involvement in the TPD activities. We will also address key issues that are specific to large-scale TPD efforts, such as compatibility of school culture, policy and management with reform efforts, and technical capabilities of teachers and school staff.

The envisioned prototype implementation is grounded in the concept of lesson case studies and offers facilitated face-to-face and online discussions for participant teachers. The infrastructure for the online program offered in years 3 and 4 will extend an existing technology-based approach such as the STEP system (see www.estep.org). Our TPD program will utilize a library of digital resources, including video cases of classroom lessons and student learning, gleaned from our classroom observations and interviews. All aspects of video case development and use will be evaluated in the course of developing this library, including: (a) planning which concepts/themes will be depicted in video cases; (b) developing procedures for editing raw classroom video to create succinct and interesting cases that emphasize important themes; (c) evaluating alternative activity structures and discussion environments that will utilize the video cases; and (d) developing a technology for storing, retrieving and presenting video cases in both face-to-face and online environments. In addition to supporting our own TPD research, we envision this digital library as a valuable general resource for supporting a range of TPD programs focused on helping students to transition from arithmetic to algebraic reasoning.

This project is designed, conducted, and interpreted by an interdisciplinary team of investigators from mathematics and mathematics education, developmental psychology, educational psychology, technology and learning environment design, teaching and teacher education, research methods and program evaluation. The investigative team represents three often-disparate research traditions: psychometrics, cognitive science, and situativity. We take a multi-disciplinary approach to wrestle with a plurality of perspectives, to identify theoretical and empirical synergies, as well as conflicting predictions and explanations, and to expand the theoretical and methodological knowledge of the members of the investigative team and the educational research community.