Cognitive Demands and Second-Language Learners: A Framework for Analyzing Mathematics Instructional Contexts

The issues involved in teaching English language learners mathematics while they are learning English pose many challenges for mathematics teachers and highlight the need to focus on language-processing issues related to teaching mathematical content. Two realistic-type problems from high-stakes tests are used to illustrate the complex interactions between culture, language, and mathematical learning. The analyses focus on aspects of the problems that potentially increase cognitive demands for second-language learners. An analytical framework is presented that is designed to enable mathematics teachers to identify critical elements in problems and the learning environment that contribute to increased cognitive demands for students of English as a second language. The framework is proposed as a cycle of teacher reflection that would extend a constructivist model of teaching to include broader linguistic, cultural, and cognitive processing issues of mathematics teaching, as well as enable teachers to develop more accurate mental models of student learning.     

The Teacher's Discourse Moves: A Framework for Analyzing Discourse in Mathematics Classrooms

In this paper a framework is proposed for analyzing the deliberate actions taken by a teacher to participate in or influence the discourse in mathematics classrooms, and such actions are referred to as the teacher's discourse moves. This work synthesizes elements of several other discourse frameworks, including those of Richards, Sfard, Cobb, and Knuth and Peressini. Expanding on the improvisational dance metaphor of Heaton's, the framework views the teacher in the additional multiple roles as a Choreographer/Stage Manager/Director of classroom discourse. Several research applications of the discourse framework to collegiate mathematics education are discussed, including discourse around collaborative problem solving in Treisman Emerging Scholars workshops, a video‐based study of a college‐level geometry course for teachers, discourse in wireless networked classrooms, and the asynchronous discourse in an online statistics course.     

A Situated and Sociocultural Perspective on Bilingual Mathematics Learners

My aim in this article is to explore 3 perspectives on bilingual mathematics learners and to consider how a situated and sociocultural perspective can inform work in this area. The 1st perspective focuses on acquisition of vocabulary, the 2nd focuses on the construction of multiple meanings across registers, and the 3rd focuses on participation in mathematical practices. The 3rd perspective is based on sociocultural and situated views of both language and mathematics learning. In 2 mathematical discussions, I illustrate how a situated and sociocultural perspective can complicate our understanding of bilingual mathematics learners and expand our view of what counts as competence in mathematical communication.     

A Situated and Sociocultural Perspective on Bilingual Mathematics Learners

My aim in this article is to explore 3 perspectives on bilingual mathematics learners and to consider how a situated and sociocultural perspective can inform work in this area. The 1st perspective focuses on acquisition of vocabulary, the 2nd focuses on the construction of multiple meanings across registers, and the 3rd focuses on participation in mathematical practices. The 3rd perspective is based on sociocultural and situated views of both language and mathematics learning. In 2 mathematical discussions, I illustrate how a situated and sociocultural perspective can complicate our understanding of bilingual mathematics learners and expand our view of what counts as competence in mathematical communication.     

Math Autobiographies: A Window into Teachers' Identities as Mathematics Learners

Mathematics autobiographies have the potential to help teachers reflect on their identities as mathematics learners and to understand their role in the development of their students' mathematics identities. This paper reports on a professional development project for K‐2 teachers (n = 41), in which participants were asked to write mathematics autobiographies. Using an adaptation of an existing framework for characterizing teachers' mathematics stories, we describe the consistencies among the participants' experiences as mathematics learners and the events that are identified as being the impetus for a transition from a negative to a positive attitude toward mathematics. Implications for both teachers and teacher educators are presented.     

Exploring Mathematics in Imaginative Places: Rethinking What Counts as Meaningful Contexts for Learning Mathematics

This paper explores what happens when students engage with mathematical tasks that make no attempt to be connected with students' everyday life experiences. The investigation draws on the work of educators who call for a broader view of what might count as real and relevant contexts for studying mathematics. It investigates students' experiences with two imaginative tasks and reports on the students' intellectual and emotional engagement. This engagement is examined and described in terms of the character and quality of the class and group discussions generated. Findings suggest that students can indeed engage productively with mathematics when it is explored in imaginative settings and that such contexts can help students support and sustain their engagement with the mathematics in the task.     

Diversifying the Curriculum in a Mathematics Classroom Streamed for High-Ability Learners: A Necessity Unassumed

Gifted learners are considerably more diverse than they are generally believed to be ( Matthews & Keating, 1995 ). Consequently, curricula must provide a wide range of learning opportunities, not simply a lock-step program at an accelerated pace. In the present study, the authors developed an experimental functions curriculum in which everyone learned a core set of concepts, then explored different kinds of functions independently using computer graphing technology. This was tested against a more typical, text-based program. Two groups of high-ability eighth-grade boys participated. Experimental students scored higher on a posttest than control students and showed much more diversity in their answers.     

Examining Instructional Practices of Elementary Science Teachers for Mathematics and Literacy Integration

Integration of content in core disciplines is viewed as an important curricular component in promoting scientific literacy. This study characterized the current practices of a group of elementary teachers relative to their development of interdisciplinary links between science, mathematics, and literacy. A qualitative analysis of survey data showed that there were substantial differences in the use of a well‐developed process for integrating instruction. Teachers also lacked a conceptual connection to integration, showed contradictions in the importance placed on hands‐on experiences, used measurement as the primary interdisciplinary connection between mathematics and science, and did not use instructional strategies designed specifically for nonfiction/expository text. The findings underscore the need for professional development that assists teachers in changing their conceptual perspectives to integration while also building pedagogical knowledge related to integration of science, mathematics, and literacy.     

A Framework for Analyzing Local Convergence Properties with Applications to Proximal-Point Algorithms

A general algorithmic scheme for solving inclusions in a Banach space is investigated in respect to its local convergence behavior. Particular emphasis is placed on applications to certain proximal-point-type algorithms in Hilbert spaces. The assumptions do not necessarily require that a solution be isolated. In this way, results existing for the case of a locally unique solution can be extended to cases with nonisolated solutions. Besides the convergence rates for the distance of the iterates to the solution set, strong convergence to a sole solution is shown as well. As one particular application of the framework, an improved convergence rate for an important case of the inexact proximal-point algorithm is derived.     

Student and Teacher Perspectives Across Mathematics and Science Classrooms: The Importance of Engaging Contexts

Substantial recent focus has been placed upon the competitiveness of American students in increasingly global economies and entrepreneurial enterprises. As concerns center on students’ educational preparedness and their efforts at continued learning, researchers acknowledge the importance of student engagement with school. In order to foster engaged learners, teachers must be able to determine and monitor their students’ levels of engagement. The current study examined the alignment of perceptions of engagement by students, teachers, and outside observers across middle and high school mathematics and science classrooms. Results indicated significant teacher‐student differences in perceptions of student cognitive engagement across mathematics and science classrooms with teachers consistently perceiving higher levels than students. Moreover, most effect sizes were moderate to large. A subsequent multi‐level analysis indicated that while teacher perceptions of student cognitive engagement were somewhat predictive of student reported cognitive engagement, academic engagement ratings by outside observers were not.     

Classroom Interactions Around Problem Contexts and Task Authenticity in Middle School Mathematics

It has been theorized that contextual tasks support student engagement and sense making. Yet, contradictory ideas exist about the role of these tasks in lessons, and further research is needed to explore how classroom interactions can help achieve their intended purposes. Through video observation of lessons in three eighth-grade classrooms using a problem-based curriculum, I investigated how teachers and students interact around problem contexts in written tasks. I found that they discussed contexts in multiple ways, falling into five general categories: referencing, positioning, elaborating, clarifying, and meta-level commentary. Using this framework, I considered how interactions around contexts related to the authenticity of tasks as written and enacted (Palm, 2006). In several lessons, these interactions led to higher authenticity as enacted than as written. These results offer a framework for interpreting context-related classroom interactions and suggest implications for instruction and research on the role contexts might play in mathematics lessons.     

Middle School Mathematics Teachers’ Perceptions of the Common Core State Standards for Mathematics and Its Impact on the Instructional Environment

This study examines Middle School Mathematics Teachers’ (MSMTs') (N = 1,241) perceptions of the Common Core State Standards for Mathematics (CCSSM) and its impact on the instructional environment. A total of eight factors appear in the data. These factors include professional support, teachers’ use of district‐adopted and non‐district‐adopted curricular resources, influence of CCSSM assessment and teacher evaluation on the instructional environment, influence of the CCSSM on classroom instruction, teachers’ planning practices, and perceived rigor of the CCSSM. The data suggest that teachers’ use of digital resources located online was disconnected from their district‐adopted resources suggesting a lack of curricular coherence. MSMTs note that the CCSSM had caused them to incorporate more reform‐oriented practices such as exploration and productive struggle into their daily instruction. MSMTs also perceive that the CCSSM includes new content that is more rigorous than previous state standards. Sampled MSMTs state that while state assessments will measure CCSSM content, they are less likely to include more complex problems or the standards for mathematical practice. Teachers are more likely to read teacher resources than student textbook activities online and to use digital resources for remediation instead of inquiry activities. Over one‐third of MSMTs wanted more CCSSM professional development.     

Design and Implementation of a Framework for Defining Integrated Mathematics and Science Education

Integrated mathematics and science teaching and learning is a widely advocated yet largely unexplored phenomenon. This study involves an examination of middle school integrated mathematics and science education from two perspectives—theory and practice. The theoretical component of this research addresses the ill-defined nature of the phrase integrated mathematics and science education. A conceptual framework in the form of a Mathematics/Science Continuum is presented to lend clarity and precision to this phrase. The theoretical framework is then used to guide analysis of tasks students are engaged in during instructional practice in middle school classrooms, where the goal of instruction is full integration of mathematics and science. Barriers to integrating mathematics and science in the school curriculum are also presented.     

A Comment on: Forecasting and Stock Control for Intermittent Demands

Journal of the Operational Research Society -