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Productive Struggle in Teaching and Learning Middle School Mathematics

by

Hiroko Kawaguchi Warshauer

Texas State University

Abstract

Mathematics educators and researchers suggest that struggling to make sense of mathematics is a necessary component of learning mathematics with understanding. This study examined students productive struggle as students worked on tasks of higher cognitive demand in middle school mathematics classrooms. Observations of 186 episodes of student-teacher interactions revealed types of struggles students encountered, the ways teachers responded to these struggles, and the kinds of interaction outcomes that were productive or not. A productive struggle framework was developed to examine the phenomenon of student struggle from initiation, interaction, to its resolution.

Introduction

Students struggle with learning mathematics is often cast in a negative light and viewed as a problem in mathematics classrooms (Hiebert & Wearne, 2003). Mathematics educators and researchers James Hiebert and Douglas Grouws suggest, however, that struggling to make sense of mathematics is a necessary component of learning mathematics with understanding (Hiebert & Grouws, 2007).

While the phenomenon we call struggle may be internal, it is also observable in most classrooms. Therefore, a portrayal of what a productive students struggle looks like set in the naturalistic setting of classroom instruction can reveal and provide insight into how aspects of teaching can support rather than hinder this instructional process which research suggests is of benefit to students understanding of mathematics (Kilpatrick, Swafford, & Findell, 2001; Hiebert & Grouws, 2007).

In order to investigate the possible connection between struggle and learning, I examined students productive struggle as students worked on tasks of higher cognitive demand in middle school mathematics classrooms. By students productive struggle, I refer to students effort to make sense of mathematics, to figure something out that is not immediately apparent (Hiebert & Grouws, 2007, p. 287), as opposed to students effort made in despair or frustration. In particular, I investigated episodes during instruction where students made mistakes, expressed misconceptions, or claimed to be lost or confused, and to which teachers responded. The kind of guidance and structure teachers provide may either facilitate or undermine the productive efforts of students struggle (Stein, Smith, Henningsen, & Silver, 2000; Doyle, 1988). A close examination of interactions in the classroom both between teacher and students and among students helped to reveal the nature of the struggles students were having in making sense of mathematics. I also observed and analyzed the features of teaching and the choices teachers made to guide the students in ways that were either productive or not productive in developing students understanding of their problem and the strategies and reasoning needed to solve it.

My study focused on the following research questions:

1. What are the kinds and patterns of students struggle that occur while students are engaged in mathematical activities that are visible to the teacher and/or apparent to the student in middle school mathematics classrooms?

2. How do teachers respond to students struggle while students are engaged in mathematical activities in the classroom?

3. What kinds of teacher responses appear to be productive in students understanding and engagement?

Conceptual Framework

The phenomenon of struggle as mentioned above refers to the intellectual effort students expend to make sense of mathematics (Hiebert & Grouws, 2007) that is challenging but reasonably within the students capabilities, possibility with some assistance. These kinds of difficulties, namely the struggles that push the students in their thinking, can play an important role in deepening students understanding if directed carefully toward a resolution (Hiebert & Grouws, 2007). My conceptual framework, therefore, is built on three main components: (1) The role of struggle in learning mathematics with understanding; (2) The nature and types of mathematical tasks and their relationship to students struggle; and (3) The ways teachers respond to students struggle in classroom interactions. Because my study about struggle is in the context of learning mathematics with understanding and the influence of teaching on the development of that understanding, it is important to consider what constitutes the nature of mathematics and what it means to engage in and be competent in the discipline (Schoenfeld, 1988). My study uses the perspective of mathematics as a social phenomenon, where people create objects and study the patterns and relationships of these objects within a social culture (Hersh, 1997; NCTM, 2000). I also take the view that mathematics is a dynamic discipline that involves exploring problems, seeking solutions, formulating ideas, making conjectures, and reasoning carefully and is not a static discipline consisting only of a structured system of facts, procedures, and concepts to be memorized or learned through repetition (Schoenfeld, 1992; Hiebert et al, 1996).

The role of struggle in learning mathematics with understanding

Researchers have looked at a variety of students attempts to make sense of mathematics that involved some difficulty: when students wrestle with problems using multiple strategies (Carpenter, Fennema, Peterson, Chiang, and Loef, 1989), undertake tasks of high cognitive demand (Stein, Grover, & Henningsen, 1996), or must explain their thinking (Hiebert and Wearne, 1993). Students from these studies showed higher levels of performance and gains in their mathematics assessments. However, not many researchers have directly studied the phenomenon of productive struggle as I have framed it; the kinds of struggle that may occur at various stages of a task when students encounter difficulty figuring out how to get started or carry out their task, are unable to piece together and explain their emerging ideas, or express an error in solving a problem.

The nature and types of mathematical tasks and their relationship to students struggle

Tasks are a central part of a teachers instructional toolkit, and what students learn is often defined by the tasks they are given (Christiansen & Walther, 1986). In order to move students toward developing a deep conceptual understanding of mathematics, classroom teaching must incorporate opportunities for students to grapple with meaningful tasks (Lampert, 2001; NCTM 1991; Schoenfeld, 1994). In addition, students must be given opportunities to make sense of important ideas in mathematics and to see connections among these ideas (Boaler & Humphreys, 2005). The students experience in the classroom of tasks of varying cognitive demand can produce different results in their learning (Hiebert & Wearne, 1993; Stein, Grover, & Henningsen, 1996). I use a task framework based on cognitive demand (Stein, Smith, Henningsen, and Silver, 2000) in order to gain a clearer picture of the kinds of tasks where these productive struggles occur.

The ways teachers respond to students struggle in classroom interaction

As well-planned as the tasks may be, students can encounter difficulty during various stages of the lesson enactment process from its introduction and development to its closure. The externalization of students struggle can engage the classroom community, or at the very least the teacher, in some response action. My conceptual framework was informed by studies that focused on interactions among the classroom participants and examined the kinds of support and guidance the interactions afforded in resolving the struggles. On the one hand, explicit actions by teachers or peers can work to build community understanding and resolve students struggle without depriving students of the opportunity to think for themselves. On the other hand, the urge by teachers to help struggling students can result in lowering or removing the cognitive demand (Henningsen & Stein, 1997) by such actions as telling students the answer (Chazan & Ball, 1999), directing the task into simpler or mechanical processes (Stein, Grover, & Henningsen, 1996), or giving guidance that funneled students thinking towards an answer without building necessary connections or meaning (HerbelEisenmann & Breyfogle, 2005).

Methodology

My investigation into the role of productive struggle in learning and teaching mathematics is exploratory in nature. The goal is to gain insight into the types and nature of student struggles that arise in middle school classrooms when students are working on tasks of higher cognitive demand and to examine the interactions that ensue between the student and teacher and the types of responses teachers use to support and resolve the student struggles. My conceptual framework suggests that learning is best supported when the teaching (1) maintains the cognitive demand of the task (2) addresses the struggle and (3) builds on students thinking. I designed my study to document these episodes of student and teacher interactions in response to student struggle, if they should arise, during enactments of tasks.

I used an embedded case study methodology (Yin, 2009) with instructional episodes as unit of analysis within the larger unit of teachers. The goal was to identify and describe the nature of the student struggles and the instructional practices of teachers that supported, guided or didnt guide the students sense-making of the mathematical tasks in the lesson episodes. I used my field notes, teacher and student interviews and episode transcripts to describe and analyze