How Mathematical Tasks Have Linked Research to Practice Over the Past 25 Years
by Margaret "Peg" Smith
For the past 25 years, Mary Kay Stein, professor and chair of the Learning Sciences and Policy program, and I—along with our graduate students and other colleagues—have been engaged in conducting research on mathematics teaching and learning and using these findings to create materials and professional education experiences for teachers.
This focus on putting research into practice, which is at the heart of the mission of the School of Education, has served to provide both a solid evidence base regarding best practices in math teaching and practical tools intended to help teachers reflect on and improve their practice. The primary focus of our work has been on the mathematical tasks that teachers use as the basis for their instruction and what students learn from these same tasks.
QUASAR (1989): All students can learn math
The story begins in 1989 with the Quantitative Understanding: Amplifying Student Achievement and Reasoning (QUASAR) project, funded by the Ford Foundation and directed by School of Education faculty member Edward Silver. The goal of QUASAR was to provide evidence that all students can learn mathematics when provided with access to high-quality instruction.
The research was conducted in six urban middle schools across the country that served economically disadvantaged neighborhoods, and focused primarily on classroom instruction and what students learned from the instruction. The key findings included the following:
Mathematical tasks with high-level demands (i.e., tasks that require thinking and reasoning rather than just the rote application of previously learned procedures) are the most difficult to implement well.
Consistent engagement with high- level tasks leads to the greatest learning gains for students.
Classroom-based factors (e.g., the questions asked, holding students accountable for thinking and reasoning, pressing for explanation and justification) shape students’ engagement with high-level tasks.
The research studies were among the first in the field to show a link between the nature of the mathematical tasks in which students engage and what they learn. The studies also highlighted the challenges faced by teachers as they seek to
enact more ambitious teaching. While the findings and methodologies inspired others in their research, we saw the results as a way to impact teachers directly.
Over the next few years, this quest to impact teaching led Stein and me to publish various practitioner articles and a book for teachers that used actual classroom examples from QUASAR to make specific suggestions for improving practice. Implementing Standards-based Mathematics Instruction: A Casebook for Professional Development, which we coauthored along with our colleagues Marjorie Henningsen (then a graduate student in the school) and Silver, provided real-life narrative cases of math instruction to help teachers to generalize key ideas about teaching and learning and then apply those ideas to their own teaching.
Subsequent work over the next decade, supported by a series of grants from the National Science Foundation, focused on:
Creating materials and professional learning experiences to help teachers select and enact high-level tasks
Conducting research to determine what teachers learn from the professional experiences in which they engage.
ASTEROID (2001): Creating successful learning experiences for teachers
The goal of A Study of Teacher Education: Research on Instructional Design (ASTEROID) was to create courses for teachers that were based on content-specific cases created through the Cases of Mathematics Instruction to Enhance Teaching project and to study both how the courses were facilitated and what teachers learned from their experiences in the courses. Two key research findings emerged from this work:
A set of practices for facilitating productive discussions of high- level tasks can be codified and learned by teachers.
Teachers can learn mathematical content and pedagogical practices through engagement in a course that included narrative cases and other practice-based materials.
This project culminated in the development of three courses intended for middle and high school teachers in proportionality, algebra, and geometry. A lesson-planning tool was created to help teachers focus on aspects of planning that research suggested would help in maintaining the demands of tasks during instruction. In addition, an article that described the five practices for orchestrating productive discussions that I coauthored with Stein was followed in 2011 by a book that provided more elaboration and has sold nearly 40,000 copies since its publication.
ESP (2003): Supporting educators to develop inquiry-oriented teaching practices
Enhancing Secondary Mathematics Teacher Preparation (ESP) focused on helping practicing teachers in the region to build their capacity for ambitious teaching through a series of workshops held over a two-year period. The more ambitious teaching in this case was inquiry oriented, meaning that students seek answers to problems, questions, or scenarios rather than simply providing established facts. The professional development in which teachers engaged in ESP was task centric and featured many of the previously developed tools and materials. Although the project was funded as a professional development effort, a research component was built in to document what contributed to teacher learning. The research yielded three key findings:
Teachers’ participating in task- centric professional development can improve their ability to enact high-level tasks in their classrooms.
Teachers’ ability to implement these high-level tasks can be sustained and improved over time.
Teachers’ learning about mathematical tasks was closely linked to the ideas represented in frameworks and their experiences in the ESP workshops.
The work on mathematical tasks continues to this day. Current research focuses on analyzing data from a schoolwide effort to impact the quality of instruction by helping teachers to collaboratively plan lessons around high-level tasks. Current material development efforts are focused on creating narrative cases to help teachers to understand the processes of reasoning and proving (i.e., looking for patterns, making conjectures, forming generalizations, and creating arguments) and how to teach them. As the work continues, new questions arise that will lead to additional cycles of research and material development.
For additional information, please see the following:
Stein, M.K., Grover, B., and Henningsen, M., “Building Student Capacity for Mathematical Thinking and Reasoning: An Analysis of Mathematical Tasks Used in Reform classrooms,” American Educational Research Journal, 33, 455–88, 1996.
Boston, M.D., and Smith, M.S., “Transforming Secondary Mathematics Teaching: Increasing the Cognitive Demands of Instructional Tasks Used in Teachers’ Classrooms,” Journal for Research in Mathematics Education, 40, 119–56, 2009.
Steele, M.D., Hillan, A.F., and Smith, M.S., “Developing Mathematical Knowledge for Teaching in a Methods Course: The Case of Function,” Journal of Mathematics Teacher Education, 16(6), 451–82, 2013.