Middle School Mathematics Classroom Research Articles

Expanding participation in problem solving in a diverse middle school mathematics classroom

In this paper, we discuss our experiences with an after-school program in which we engaged middle-school students with low socioeconomic status from an urban community in mathematical problem solving. We document that these students participated in many aspects of problem solving, including the posing of problems, constructing justifications, developing and implementing problem-solving heuristics and strategies, and understanding and evaluating the solutions of others. We then delineate what aspects of our environment encouraged the students to take part in these activities, particularly emphasising the proactive role of the teacher, the tasks the students completed, and the social norms of our after-school sessions. Finally, we discuss the relationship between our study and the literature on equity research in mathematics education.

Developing a Caring Ethic for Middle School Mathematics Classrooms

Angiline (first author) remembers her first day teaching middle school mathematics. She recalls feeling academically prepared, however, she was not ready for what was to come. She remembers being excited and enthusiastic about her first year of teaching, but she was not prepared emotionally for the 130 eighth graders she taught every day. They had so many needs. They wanted her to sign forms to go on field trips and to excuse their absences. They needed pencils, paper, and books. Mostly they wanted her attention, but Angiline had no idea how to meet the emotional needs of these young adolescents. A colleague gave Angiline a book about how to make the most of her mathematics class time. Out of desperation and exhaustion, she tried some of the suggestions to make the beginning of class a valuable time for her and her students. She began to start classes with a critical thinking exercise. Initially, Angiline did this warm-up activity to help keep the students busy while she managed administrative tasks such as taking roll, returning homework, and attending to announcements on the public address system. But as Angiline got better at designing these starting tasks, her students came to enjoy the challenging exercises, and a caring classroom community began to form. A call to care Like Angiline, teachers assigned to entry-level or low track math classes are entrusted with the students who need them the most. These students may be academically disengaged and, perhaps, do not see how education, especially mathematics, relates to their lives. Students in these classes have many labels, including at-risk, English language learners (ELL), and urban students. Students of color, especially African Americans and Latinos, are likely to be enrolled in these entry-level classes in higher proportions (Oakes, 1995). Mathematics educators are called to engage and educate every one of their students, including these disenfranchised students. Based on recommendations in the literature and their experiences in the classroom, the authors believe that for every student to be successful, teachers must foster a caring ethic (Ladson-Billings, 2006). The authors, Angiline and Al, are university teacher educators who share similar experiences building community in middle grades mathematics classrooms. In this article, they describe what a caring ethic looks like in a middle grades mathematics class, and they suggest specific practices drawn from their experiences that teachers can implement to make their classrooms more caring and engaging places. Caring community of learners As eighth grade mathematics teachers, the authors began the year building relationships with their students and encouraging communication to establish a caring ethic. In their mathematics classes, a caring community attitude was everything, and, as in any community, communication and relationships were paramount. Following Noddings (2002), an ethic of caring in the classroom has four key components. Teachers must model acting as caregivers and people who care for others in the classroom. Teachers and their students must engage in dialogue about caring as they seek to arrive at common understandings of the concept. Students must have opportunities to practice caring for other members of the classroom community. Finally, teachers must offer confirmation of the struggles and accomplishments of their students as they care for one another. Caldwell and Sholtis (2008) provided further insight into developing an ethic of care in the classroom. They identified five characteristics of student-oriented teachers who care for their students: 1. They treat all students with respect by using their names and model how to show respect for other students and other teachers. 2. Student-oriented teachers encourage positive discourse in the classroom, which they demonstrate by giving constructive written encouragement on assignments and frequent verbal praise (Caldwell, 1999). …

Regulation of Teacher Elicitations in the Mathematics Classroom

Using the perspective of instructional conversation, we investigated how one teacher regulated student participation and conceptual reasoning in the middle-school mathematics classroom. We examined the elicitations—questions and provocative statements—made by the teacher over a four-day algebra lesson. Analyses showed how the teacher systematically regulated the level of cognitive complexity of his elicitations in reaction to students' responses. When students gave inaccurate or incomplete answers, the teacher tended to reduce the level of cognitive complexity needed to respond to a subsequent elicitation, with the apparent impact being that he scaffolded participation and reasoning. When students provided responses that were mathematically accurate, the teacher usually increased the elicitation level, which subsequently engaged students in more sophisticated forms of reasoning.

Origami Instruction in the Middle School Mathematics Classroom: Its Impact on Spatial Visualization and Geometry Knowledge of Students

Within the study of geometry in the middle school curriculum is the natural development of students’ spatial visualization, the ability to visualize two- and three-dimensional objects. The national mathematics standards call specifically for the development of such skills through hands-on experiences. A commonly accepted method is through the instruction of Origami, the art of paper folding. This study focused on Origami’s impact as a teaching tool in the middle school mathematics classroom. The effects of Origami instruction on a group of seventh grade mathematics students’ (n = 56) spatial visualization skills and level of geometry understanding were investigated using a pre-test/post-test quasiexperimental design. A 2 (group) × 2 (gender) factorial method was used on gathered data via three separate spatial tests (Card Rotation, Paper Folding, and Surface Development Tests) and a subset of released National Assessment of Educational Progress questions. After controlling for initial differences, an analysis of covariance revealed a significant interaction effect between group and gender for one of three spatial visualization tests. For geometry knowledge, no significant differences were found. Results imply that Origami lessons blended within mathematics instruction are as beneficial as traditional instruction in building an understanding of geometric terms and concepts, though the approach affects the spatial ability of males and females differently.

Constructing competence: an analysis of student participation in the activity systems of mathematics classrooms

This paper investigates the construction of systems of competence in two middle school mathematics classrooms. Drawing on analyses of discourse from videotaped classroom sessions, this paper documents the ways that agency and accountability were distributed in the classrooms through interactions between the teachers and students as they worked on mathematical content. In doing so, we problematize the assumption that competencies are simply attributes of individuals that can be externally defined. Instead, we propose a concept of individual competence as an attribute of a person's participation in an activity system such as a classroom. In this perspective, what counts as “competent” gets constructed in particular classrooms, and can therefore look very different from setting to setting. The implications of the ways that competence can be defined are discussed in terms of future research and equitable learning outcomes.

Expanding the Disciplinary Expertise of a Middle School Mathematics Classroom: Re-Contextualizing Student Models in Conversations With Visiting Specialists

This article examines how conversations during design reviews in which 8th-grade mathematics students shared population models with visiting specialists expanded the disciplinary expertise of the classroom. Re-contextualizing is a conversational exchange that visiting specialists initiated to invite groups to consider their models in novel contexts. Analysis of 14 design reviews in 2 classrooms showed that re-contextualizing resulted in both the elaboration of ideas students already understood and new contributions to students' understandings of mathematical aspects of population modeling. This article presents case studies of 2 groups that differed in terms of their interest in the curricular task and the level of conceptual integrity of their population models. Despite these differences, the re-contextualizing exchanges that emerged in their design reviews led to new insights for both groups and provided them with opportunities to try on ways of thinking and acting like population biologists.

The fragility of group flow: The experiences of two small groups in a middle school mathematics classroom

This article considers two small groups of students in the same Grade 8 mathematics classroom whose approaches to the same mathematical problem result in very different experiences. Using videotapes and written transcripts, an analysis of the groups’ working processes was undertaken using Sawyer's pre-existing structures required for the presence of group flow, and Davis and Simmt's conditions for complex systems. It is suggested that although both groups had the prerequisite structures in place to experience group flow, the second group was not decentralized enough to enable all members to establish a working collaborative proximal zone of development in which they could develop their ideas as a collective, while the first group was sufficiently decentralized and appeared to demonstrate episodes of experiencing group flow. If teachers are aware of conditions that encourage the experience of group flow, this may help them in forming productive small groups within the classroom and developing successful group-oriented learning tasks.

To Disagree, We Must Also Agree: How Intersubjectivity Structures and Perpetuates Discourse in a Mathematics Classroom

Learning in a socially mediated context like a classroom places emphasis on the ability of learners to communicate their ideas to others, and for members of a class to achieve shared meaning or intersubjectivity (IS). We take a participatory view of IS, where both consensual agreement and disagreement are regarded as aspects of a common set of processes that mediate collective activity. Interlocutors need not demonstrate convergence toward a common idea or solution to exhibit IS and, indeed, they appear to need a shared understanding to express substantive disagreement through divergent views. Multilevel, multimodal analyses of videotape of a middle school mathematics classroom, including speech, gestures, drawing, and object use, reveal a discourse that is organized into recurrent sequences of event triads. The dynamics toward and away from convergent ideas appears to be instrumental in fostering sustained and engaging discourse and influencing the representations that students propose during problem solving. Participants frequently exhibited IS, but, as allowed for in the participatory view, the interactions did not seem to convert many students from their initial interpretations. Instead, disagreements and a desire to establish common understanding appeared to lead participants to express their divergent views in more refined and accessible ways. Advancement of our understanding of the role that IS serves in socially mediated learning has the potential to inform both educational theory and emerging areas in embodied cognition and cognitive neuroscience that addresses imitation and empathy, and thus help to bridge research between brain function and social cognition.

How Does Your Mathematical Garden Grow?

Imagine a middle school mathematics classroom where students are actively engaged in a real-life application problem incorporating multiple Process and Content Standards as outlined by NCTM (2000). Sounds of mathematical communication arise as students use multiple representations to help connect mathematical ideas throughout the problem-solving process. Students apply various types of reasoning and explore alternate methods of proof while working attentively on applications that incorporate Number and Operations, Algebra, Geometry, and Measurement.

Teaching mathematics to middle school students with learning difficulties

Montague, Jitendra, Introduction. Bryant, Kim, Hartman, Bryant, Standards-based Mathematics Instruction and Teaching Middle School Students with Mathematical Disabilities. Woodward, Making Reform-based Mathematics Work for Academically Low-achieving Middle School Students. Xin, Jitendra, Teaching Problem-solving Skills to Middle School Students with Learning Difficulties: Schema-based Strategy Instruction. van Garderen, Teaching Visual Representation for Mathematics Problem Solving. Montague, Self-regulation Strategies for Better Math Performance in Middle School. Foegen, Evaluating Instructional Effectiveness: Tools and Strategies for Monitoring Student Progress. Kinder, Stein, Quality Mathematics Programs for Students with Disabilities. Parmar, DeSimone, Facilitating Teacher Collaboration in Middle School Mathematics Classrooms with Special-Needs Students.

“And That's Just how it Starts”: Teaching Mathematics and Developing Student Agency

This article reports on a two-year qualitative, practitioner-research study of teaching and learning for social justice. The site was my middle-school mathematics classroom in a Chicago public school in a Latino/a community. A major pedagogical goal was to create conditions for students to develop agency, a sense of themselves as subjects in the world. My research suggests that students learned mathematics and began to develop sociopolitical awareness and see themselves as possible actors in society through using mathematics to understand social injustices. This research contributes to our understanding of how to create opportunities for students to develop agency in K-12 mathematics classrooms, and may also contribute to our knowledge of developing agency in any subject area.

Code talk: Student discourse and participation with networked handhelds

This study explores the potential of networked handheld computers to support collaborative problem solving in small groups. Drawing on data from a middle school mathematics classroom equipped with a wireless handheld network, I argue that the sharing of mathematical objects through interactive devices broadens the ‘bandwidth’ of classroom collaboration, expanding the range of participatory forms through which students might contribute to the work of a group and enhance their own learning. The analysis focuses on the participation strategies of those students in two focus groups who were most able to demonstrate posttest score gains from relatively low scores on a pretest. In particular, the device network provided those students with a set of collective, dynamic objects through which they supplemented and coordinated discursive forms of participation in the joint work of their respective groups.

Small-Group Reflections: Parallels Between Teacher Discourse and Student Behavior in Peer-Directed Groups

Prior research on small-group collaboration identifies several behaviors that significantly predict student learning, such as exchanging explanations and applying help received. Previous reports focus on student behavior to understand why many students do not engage in behaviors predictive of learning, leaving unexplored how teachers may influence small-group interaction. Consequently, this article describes established teacher practices and examines whether and how students reproduce teacher discourse in the context of a semester-long program of cooperative learning in four middle school mathematics classrooms. This article illustrates changes that emerged and factors that made change difficult. The research concludes that student behavior largely mirrored the discourse modeled by and the expectations communicated by teachers. Teachers tended to give unlabeled calculations, procedures, or answers instead of labeled explanations. Teachers often instructed using a recitation approach in which they assumed primary responsibility for solving the problem, having students only provide answers to discrete steps. Finally, teachers rarely encouraged students to verbalize their thinking or to ask questions. Students adopting the role of help-giver showed behavior very similar to that of the teacher: doing most of the work, providing mostly low-level help, and infrequently monitoring other students' level of understanding. The relatively passive behavior of students needing help corresponded to expectations communicated by the teacher about the learner as a fairly passive recipient of the teacher's transmitted knowledge.

Standards-based mathematics assessment in middle school: rethinking classroom practice

This volume takes an in-depth look at the problems and practices involved in conducting formative assessments in middle school mathematics classrooms. Researchers and teachers identify the challenges teachers faced as they attempted to implement new assessment procedures.

You Read Me a Story, I Will Read You a Pattern

Have You Ever Wondered how to use children's literature in a middle school mathematics classroom? In today's standards-driven environment, aligning activities to various standards is important. Children's books can be the perfect introduction to a unit or lesson. Paying careful attention to the elements in the story and using a little imagination, creativity, and a working knowledge of the mathematics Standards are all the items needed to begin.

Developing Mathematical Thinking and Self-Regulated Learning: A Teaching Experiment in a Seventh-Grade Mathematics Classroom.

Mathematics educators have found sociocultural models of teaching and learning to be powerful in their ability to describe and support the pursuit of instruction based on recent standards documents (e.g., National Council of Teachers of Mathematics [NCTM], 1989,2000). These models of instruction,however, have been criticized for their lack of explicitness. Detailed descriptions of cognitive processes within self-regulated learning (SRL) and attribution theories (e.g., Borkowski etal., 1988; Butler, 2002; Zimmerman, 2000)lend support for and provide examples of explicit instruction embedded within sociocultural models of mathematics instruction. Self-regulated learners are active participants in their own learning,are able to select from a repertoire of strategies and to monitor their progress in using these strategies toward a goal. In this article, we describe the collaborative efforts of a seventh-grade mathematics teacher and a university faculty member to develop students' mathematical thinking and self-regulation within one middle school classroom. We argue that implementing mathematics instruction consonant with NCTM(1989, 2000) standards documents makes the development of self-regulated learners possible and is at the same time dependent upon some degree of self-regulation within the community of practice. Several factors are crucial to this development: multiple representations and rich mathematic al tasks; classroom discourse; environmental scaffolding of strategic behavior; and varying needs for explicitness and support. This work constitutes beginning efforts to describe contexts that support the development of mathematical thinking and SRL in the middle school mathematics classroom.

Middle School Mathematics Classrooms: A Place for Latina Parents' Involvement

The study is of 3 Latina parents of children in an Arizona middle school in which parents made observations and participated in three different reform mathematics classrooms. The parents' expectations, concerns, and reactions to the observations of the lessons were discussed and recorded. The goal of the study was to better understand the Latina parents' perspectives about their children's mathematics education. The findings from this study can be used to guide teachers' and parents' efforts to develop effective programs for educating parents about mathematics education reform.

Dilemmas of Discourse-Oriented Teaching in One Middle School Mathematics Classroom

In this article we focus on the middle school mathematics lessons of 1 teacher who has been relatively successful at instituting discourse-oriented teaching in her mathematics classes over a 3-year...

Implementing Data Analysis in a Sixth-Grade Classroom

A middle school mathematics classroom is an exciting place to be. The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) presents for grades 5-8 many curricular idea that have generated much enthusiasm by incorporating teaching strategies that promote active student learning. Middle school teachers have taken great strides toward implementing ideas that are both innovative and instructive. This article explains how one teacher is converting her traditional mathematics classroom into one that more closely reflects the standards document. These result could not have happened as quickly without the cooperative efforts of school and university personnel.

Linking Instruction and Assessment in a Middle School Mathematics Classroom

This statement, made by a sixth grade student at the conclusion of her working on a performance-based mathematics activity, captures an important feature of such an activity—that it is a realistic problem which holds meaning within the student's under standing of the world. Rather than math ematics problems being presented as small-size, isolated (why this?) type of exercises, high quality performance activi ties require the students to use their ma thematic skills and knowledge in a larger, applied real-life context. The activity completed by this sixth grader is part of the PACKETSTM Program for Middle School Mathematics, a research based performance assessment program cur rently under developmentby the Educational Testing Service (ETS). The PACKETS Pro gram is a set of classroom-based materials for teachers to use as part of their instruc tional program. The activities are designed in such a way that student learning is pro moted at the same time that the teacher is