Mathematics Instructional Practices Research Articles

Teachers’ perceptions of examining students’ thinking: Changing mathematics instructional practice

AbstractThis paper seeks to illuminate teachers’ perceptions of the challenges and benefits of systematically examining students’ thinking as part of a professional development program in elementary mathematics education. Using a framework of models of conceptual change and principles of discomfort, three elementary teachers’ perceptions of their experiences with reform-oriented mathematics instruction and systematically examining students’ thinking were analyzed. Analysis of interview data yielded five organizing themes. Findings offer insight into understanding the efficacy of examining students’ thinking as a professional development strategy.

Which Instructional Practices Most Help First Grade Students with and without Mathematics Difficulties?

We used population-based, longitudinal data to investigate the relation between mathematics instructional practices used by 1st grade teachers in the U.S. and the mathematics achievement of their students. Factor analysis identified four types of instructional activities (i.e., teacher-directed, student-centered, manipulatives/calculators, movement/music) and eight types of specific skills taught (e.g., adding two-digit numbers). First-grade students were then classified into five groups on the basis of their fall and/or spring of kindergarten mathematics achievement-three groups with mathematics difficulties (MD) and two without MD. Regression analysis indicated that a higher percentage of MD students in 1st grade classrooms was associated with greater use by teachers of manipulatives/calculators and movement/music to teach mathematics. Yet follow-up analysis for each of the MD and non-MD groups indicated that only teacher-directed instruction was significantly associated with the achievement of students with MD (covariate-adjusted ESs = .05-.07). The largest predicted effect for a specific instructional practice was for routine practice and drill. In contrast, for both groups of non-MD students, teacher-directed and student-centered instruction had approximately equal, statistically significant positive predicted effects (covariate-adjusted ESs = .03-.04). First-grade teachers in the U.S. may need to increase their use of teacher-directed instruction if they are to raise the mathematics achievement of students with MD.

Relationship between Teacher Efficacy and Self-Reported Instructional Practices: An Examination of Five Asian Countries/Regions Using TIMSS 2011 Data.

Teacher efficacy in a particular content area is seen as an important factor shaping teaching practice and student learning in the Western literature. However, inadequate efforts have been made to examine this assumption empirically. Drawing on the Trends of International Mathematics and Science Study (TIMSS) 2011 mathematics teachers’ data at the eighth grade level, this study examines the relationship between teacher efficacy in teaching mathematics and teachers’ mathematical instructional practices in five Asian countries/regions. Guided with Bandura’s social cognitive theory, this study finds that overall teachers in the five countries/regions reported much lower teacher efficacy in teaching mathematics and lower levels of instructional practices than the international norms. The relationship between teacher efficacy and instructional practices varied within these countries/regions. Teacher efficacies in teaching mathematics are not always statistically and positively related to their instructional practices in engaging students. Together, these findings challenge the theoretical assumption about the positive relationship between teacher efficacy and instructional practices. Possible reasons of these findings and future directions are discussed.

Quality Mathematics Instructional Practices Contributing to Student Achievements in Five High-Achieving Asian Education Systems: An Analysis Using TIMSS 2011 Data

Although teaching quality is seen as crucial in affecting students’ performance, what types of instructional practices constitute quality teaching remains a question. With the theoretical assumptions of conceptual and procedural mathematics teaching as a guide, this study examined the types of quality mathematics instructional practices that affect students’ mathematics learning across five high-performing Asian education systems using the Trends in International Mathematics and Science Study (TIMSS) 2011 dataset. It found that no combinations of the components of conceptual and procedural mathematics teaching practices exist consistently across the five education systems. Results from the study provide important implications for practitioners and policy makers regarding how to improve mathematics teaching and learning in these Asian education systems as well as elsewhere.

Quality Mathematics Instructional Practices Contributing to Student Achievements in Five High-Achieving Asian Education Systems: An Analysis Using TIMSS 2011 Data

Although teaching quality is seen as crucial in affecting students’ performance, what types of instructional practices constitute quality teaching remains a question. With the theoretical assumptions of conceptual and procedural mathematics teaching as a guide, this study examined the types of quality mathematics instructional practices that affect students’ mathematics learning across five high-performing Asian education systems using the Trends in International Mathematics and Science Study (TIMSS) 2011 dataset. It found that no combinations of the components of conceptual and procedural mathematics teaching practices exist consistently across the five education systems. Results from the study provide important implications for practitioners and policy makers regarding how to improve mathematics teaching and learning in these Asian education systems as well as elsewhere.

Examining the Relationship Between Teachers’ Instructional Practices and Students’ Mathematics Achievement

The purpose of this study was to determine whether relationships existed between teachers’ implementation of two specific discourse-related instructional practices and students’ mathematics achievement in geometry and measurement as part of a research study on the effectiveness of an advanced mathematics curriculum for kindergarten and Grades 1 and 2. The mathematics units incorporated the following instructional practices: engaging students in verbal communication in mathematics and encouraging the use of appropriate mathematical vocabulary. Hierarchical linear modeling was used to determine the relationships between teachers’ use of the instructional practices and the students’ mathematics achievement. Results indicated that significant, positive relationships existed; the teachers’ implementation scores for the verbal communication and encouraging mathematical language instructional practices were predictors of student mathematics achievement as measured by students’ percentage gain scores on the Open-Response Assessments. Implications of these findings for mathematics instruction are discussed.

Evaluating the structure of the Mathematics Teacher Questionnaire: A measure of exposure to mathematics instructional practices and content

Psychometric properties of 24 items from the fifth grade Early Childhood Longitudinal Study-Kindergarten Cohort Mathematics Teacher Questionnaire were investigated in a sample of 5,181 participants. These items asked teachers to report how often they had their classroom students engage in different mathematics content, skills and instructional practices. Exploratory factor analyses were employed to evaluate the underlying factor structure of the selected items. Results suggest that these items effectively measure three distinct latent constructs that are specific to mathematics instruction: Instructional Practices for Teaching Mathematics, Developing Number Sense, and Beyond Number Sense. These results support the use of this instrument as a guide for research and policy, or to inform teacher curriculum decisions.

Mathematical Habits of Mind for Teaching: Using Language in Algebra Classrooms

ABSTRACT:The notion of mathematical knowledge for teaching has been studied by many researchers, especially at the elementary grades. Our understandings of this notion parallel much of what we have read in the literature, but are based on our particular experiences over the past 20 years, as mathematicians engaged in doing mathematics with secondary teachers. As part of the work of Focus on Mathematics, Phase II MSP, we are developing, in collaboration with others in the field, a research program with the ultimate goal of understanding the connections between secondary teachers' mathematical knowledge for teaching and secondary students' mathematical understanding and achievement. We are in the early stages of a focused research study investigating the research question: What are the mathematical habits of mind that high school teachers use in their professional lives and how can we measure them? The main focus of this paper is the discussion of the habit of using mathematical language, and particularly how this habit plays out in a classroom setting.Keywords: Mathematical habits of mind, mathematical language, algebra(ProQuest: ... denotes formulae omitted.)Our Philosophy and ApproachBuilding on two decades of prior work, the Focus on Mathematics (FoM] Math and Science Partnership program (MSP) has, over the last decade, developed and refined a distinctive framework for a mathematics-centered approach to developing teacher leaders, and it has built a mathematical community based on that framework. The FoM approach involves teachers, mathematicians, and educators working together in professional development activities. The common thread running through this tightly connected set of activities is an explicit focus on mathematical habits of mind.We define mathematical habits of mind (MHoM] to be the web of specialized ways of approaching mathematical problems and thinking about mathematical concepts that resemble the ways employed by mathematicians (Cuoco, Goldenberg, & Mark, 1997, 2010; Goldenberg, Mark, & Cuoco, 2010; Mark, Cuoco, Goldenberg, & Sword, 2010}. These habits are not about particular definitions, theorems, or algorithms that one might find in a textbook; instead, they are about the thinking, mental habits, and research techniques that mathematicians employ to develop such definitions, theorems, or algorithms. Some examples of MHoM follow:* Discovering the structure that is not apparent at first by experimenting and seeking regularity and/or coherence.* Choosing a useful representation-or purposefully toggling among various representations-of a mathematical concept or object.* Purposefully transforming and/or interpreting algebraic expressions (e.g., rewriting x^sup 2^ -6.x+ 10 as (;t-3)^sup 2^ + l to reveal its minimum value].* Using mathematical language to express ideas, assumptions, observations, definitions, or conjectures.Our work over the past decade has convinced us of the importance of MHoM for students and for teachers of mathematics, particularly at the secondary level. These habits foster the development and use of general purpose tools that make connections among various topics and techniques of secondary school mathematics content; they can bring parsimony, focus, and coherence to teachers' mathematical thinking and, in turn, to their work with students. In this sense, we envision MHoM as a critical component of mathematical knowledge for teaching (Hill, Rowan & Ball, 2005) at the secondary level (i.e., the knowledge necessary to carry out the work of teaching mathematics).We begin this paper by describing the mathematical community that we have built and the impact that it has had on our teachers, in particular, the impact on teachers' mathematical understanding and instructional practices. Then we discuss the research that grew out of our desire to study scientifically how MHoM might be an indicator of teacher effectiveness. …

Classroom Instruction and the Mathematics Achievement of Non-English Learners and English Learners

ABSTRACT The authors, in a nonexperimental randomized study, used national data from the Early Childhood Longitudinal Study–Kindergarten Cohort (ECLS-K) to examine present instructional practices for Grade 5 mathematics classrooms and its impact on achievement for White non-Hispanic non–English language learners (ELLs), Hispanic non-ELLs, and Hispanic English language learners. The following research questions were addressed: (a) Were there significant differences in mathematics instructional practices among White non-Hispanic non-ELLs, Hispanic non-ELLs, and Hispanic-ELLs? (b) What were the effects of the mathematical instructional practices on fifth-grade mathematics achievement for these groups? Findings reveal that students’ mathematics achievement in previous grades directly impacted students’ achievement in future grades. Findings also demonstrated limited variance in the instructional practices used by teachers.

Investigating the Relationship between Elementary Teacher Mathematics Anxiety, Mathematics Instructional Practices, and Student Mathematics Achievement

Many elementary teachers have been found to have high levels of Many elementary teachers have been found to have high levels of mathematics anxiety but the impact on student achievement was unknown. Elementary teachers ( N = 692) completed the modified Mathematics Anxiety Rating Scale-Revised (Hopko, 2003) along with a questionnaire probing anxiety about teaching mathematics and current mathematics instructional practices. Student mathematics achievement data were collected for the classrooms taught by the teachers. A positive relationship was found between anxiety about mathematics and anxiety about teaching mathematics. Increased student mathematics achievement was related to lower levels of anxiety about teaching mathematics, but was not related to general anxiety about mathematics. Anxiety about mathematics was positively related to more National Council of Teachers of Mathematics (NCTM) Standards-based instructional practices.

Action research improves math instruction

Three Suffolk, Virginia, elementary school teachers—Paula Beckett, Deb McIntosh, and Leigh-Ann Byrd—are also new graduate-school candidates who have enrolled in the mathematics specialist program at their local university. Upon learning that an action research class was one of their first requirements, all were feeling a bit intimidated and quite overwhelmed. That soon changed as they began the “practitioner-based” research process in their math classrooms. Beckett, McIntosh, and Byrd share their journeys of self-study and the spirit of critically examining their own mathematics instructional practices. Their discussions highlight the research projects and illustrate how teachers can take charge of their development as well as investigate issues related to student achievement within the confines of their classrooms.

Technology, Mathematics PS/RTI, and Students with LD: What Do We Know, What Have We Tried, and What Can We Do to Improve Outcomes Now and in the Future?

Effective integration of technology within mathematics problem solving/response to instruction (PS/RTI) for students with/at risk for identification of learning disabilities (LD) is the focus of this article. The discussion is situated within the current context of mathematics PS/RTI, the literature base relative to technology integration in mathematics for students with LD, and emerging 21st-century technologies. Limitations of the research base are identified, and suggestions for the future are provided. Next, a systematic framework for integration of technology in mathematics PS/RTI is introduced that incorporates consideration of (a) effective mathematics instructional practice within mathematics PS/RTI, and (b) the nature of the mathematics content and student learning barriers and strengths. Ways in which technology can enhance effective mathematics practices for students with LD are suggested, and implications for research, development, and practice are discussed.

Mathematics Teachers' Professional Development Opportunities in the Curriculum Reform of China

This study examined the professional development opportunities that about eighty mathematics teachers of junior high schools were provided with in the national curriculum reform of China and explored how their beliefs in mathematics instruction and instructional practice were affected by the professional development opportunities available to them. A questionnaire was administered to collect data. The results showed that the length of time the participants spent on attending school-based learning activities had a significant effect on their attitudes toward the effectiveness of the professional development and on their instructional practice. The content of the training programs that focused on studying textbooks and observing and reflecting upon model lessons was highly correlated to the implementation of conventional mathematics instruction. Another factor that related to the implementation of conventional mathematics instruction was the collegial support the teachers received at their schools.

Comparing Self-paced and Cohort-based Online Courses for Teachers

The study investigated whether online professional development courses with different levels of support have different impacts on teacher outcomes. Variations of an online course for middle school algebra teachers were created for four experimental conditions. One was a highly supported condition, with a math education instructor, an online facilitator, and asynchronous peer interactions among participants available as participants worked through the course together. Another was a self paced condition, in which none of the supports were available. The other two conditions provided intermediate levels of support. All conditions showed significant impact on teachers’ mathematical understanding, pedagogical beliefs, and instructional practices. Surprisingly, the positive outcomes were comparable across all four conditions. Further research is needed to determine whether this finding is limited to self selected teachers, the specifics of this online course, or other factors that limit generalizability.

The Impact of an Ongoing Professional Development Program on Prekindergarten Teachers' Mathematics Practices

Mathematics is a natural part of daily life for young children as they explore and investigate the world around them. To build on these experiences, and to begin establishing a mathematical foundation, early childhood educators must not only be knowledgeable about mathematical concepts, they must also be aware of the most developmentally appropriate ways in which to teach these concepts to young children. After participation in an ongoing professional development program, specifically targeting teachers of prekindergarten children in public school, Preschool Programs for Children with Disabilities (PPCD), Head Start, and child care settings, teachers reported positive changes in math practices. Specifically, teachers reported a stronger alignment to national mathematics standards and increased awareness pertaining to developmentally appropriate mathematics practices as they apply to early childhood classrooms. Teachers reported a shift towards more hands-on activities and a shift away from the use of worksheets in their prekindergarten classrooms. Implications from this study suggest that ongoing professional development that is designed to meet the specific needs of early childhood educators can have a positive impact on reported mathematics content knowledge and instructional practices.

Rejoinder to the Critiques of the National Mathematics Advisory Panel Final Report

The authors respond to comments and criticisms made of Foundations for Success: The Final Report of the National Mathematics Advisory Panel (2008). Given the requirements of the president’s executive order, constraints of time, and the need to make recommendations generalizable to national policy, the Panel adopted appropriately high evidentiary standards for its recommendations. The authors also respond to criticisms voiced on Panel findings regarding mathematics curricula, instructional practices, learning processes, teachers and teacher education, and assessment. While scholars should continue to develop promising programs of research, experimental or otherwise, the report offers a solid basis for improving math achievement.

Preservice Teachers’ Disposition Self-Appraisals: Is There a Connection to Mathematics Instructional Practices?

This paper explores preservice teachers’ dispositions and the connection to mathematics instructional practices. The Theory of Culturally Relevant Pedagogy served as the theoretical grounding for the study. Hence, its three broad propositions: conceptions of self and others, social relations, and conceptions of knowledge structure the argument presented. The participants in the study are elementary preservice teachers enrolled in a post-baccalaureate alternative certification program. The full-time, accelerated program focuses on and is specifically designed for individuals committed to teaching in urban, high-poverty schools. A dispositions survey, open-ended narratives, and teaching observation rubric served as data sources for the qualitative study. Findings indicate a connection between dispositions and instructional practices; however, nuances existed between the components of culturally relevant pedagogy.

Reconsidering the Study of Mathematics Instructional Practices: The Importance of Curricular Context in Understanding Local and Global Teacher Change

This paper discusses the case of one teacher, Jackie, whose instructional practices illuminate the importance of textbooks and student/parent expectations in shaping pedagogy. Jackie teaches in the Plainview district, which offers parents and students a choice between a reform-oriented, integrated curriculum (Core Plus) and a more conventional algebra sequence (the University of Chicago series). Each day, Jackie teaches two very different sections of accelerated eighth-grade mathematics using each of these curricular materials. Drawing from students’ survey responses, classroom observations, and teacher interview data, we show ways in which Jackie’s pedagogy differs considerably between the two courses and we shed light on reasons underlying this variation.

Using Structured Classroom Vignettes to Measure Instructional Practices in Mathematics

Large-scale educational studies frequently require accurate descriptions of classroom practices to judge implementation and impact. However, it can be difficult to obtain these descriptions in a timely, efficient manner. To address this problem, the authors developed a vignette-based measure of one aspect of mathematics instructional practice, reform-oriented instruction. Teachers read contextualized descriptions of teaching practices that varied in terms of reform-oriented instruction, and rated the degree to which the options corresponded to their own likely behaviors. Responses from 80 fourth-grade teachers yielded fairly consistent responses across two parallel vignettes and moderate correlations with other scales of reform-oriented instruction derived from classroom observations, surveys, and logs. The results suggested that the vignettes measure important aspects of reform-oriented instruction that are not captured by other measurement methods. Based on this work, it appears that vignettes can be a useful tool for research on instructional practice, but cognitive interviews with participating teachers provided insight into possible improvements to the items.

The mathematical tale of two teachers: A longitudinal study relating mathematics instructional practices to level of intellectual development

In this article the development of two teachers as they make the transition from pre-service teachers to experienced teachers is examined. While these teachers participated in the same mathematics methods course and similar collaborative environments in their practicum experiences, their mathematics classroom instructional practices revealed stark differences by the time they were experienced teachers. In an effort to investigate these differences, the teachers’ beliefs were explored in relation to those promoted in their pre-service course and in relation to their implementation of current mathematics education reform practices.