From the Written to the Enacted Curricula: The Intermediary Role of Middle School Mathematics Teachers in Shaping Students' Opportunity to Learn

In this paper is reported the extent of textbook use by 39 middle school mathematics teachers in six states, 17 utilizing a textbook series developed with funding from the National Science Foundation (NSF‐funded) and 22 using textbooks developed by commercial publishers (publisher‐generated). Results indicate that both sets of teachers placed significantly higher emphasis on Number and Operation, often at the expense of other content strands. Location of topics within a textbook represented an oversimplified explanation of what mathematics gets taught or omitted. Most teachers using an NSF‐funded curriculum taught content intended for students in a different (lower) grade, and both sets of teachers supplemented with skill‐building and “practice” worksheets. Implications for documenting teachers' “fidelity of implementation” ( National Research Council, 2004 ) are offered.     

Exploring the quality of the mathematical tasks in the new Turkish elementary school mathematics curriculum guidebook: the case of algebra

To take its due place in the world of education, Turkey has been through serious reform initiatives in the curriculums of various school subjects since 2003. The new Turkish elementary school curriculum was prepared considering the research studies conducted in Turkey and in other countries, as well as the educational systems of developed countries and previous experiences with mathematics education in Turkey. This study attempts to provide a perspective on the nature of the instructional tasks in the new elementary school mathematics curriculum. In particular, our focus is to explore the level of cognitive demands (LCD) in the algebra tasks provided in the national elementary mathematics curriculum guidebook. This curriculum document is a major resource for administrators, stakeholders, textbook publishers and ultimately for teachers. For every learning objective, it provides sample tasks to be used in mathematics instructions. In this study, our purpose is to explore the LCD of each of these tasks by utilizing a framework developed by Smith and Stein (Math Teach Middle School 3:344–350, 1998). The framework classifies mathematical tasks according to the level of demands: lower-level and higher-level demands. While the lower-level demands are related to memorization and procedures without connections, the higher-level demands are related to procedures with connections and doing mathematics. The findings revealed that 60% of algebra tasks for each grade level required higher LCD and a great majority of the remaining tasks were at the level of procedures without connections. The findings of the study particularly inform curriculum developers about issues regarding the quality of the tasks given in the curriculum guide and provide possible suggestions to improve the implementation of the curriculum change process.     

The quality of non-textual elements in mathematics textbooks: an exploratory comparison between South Korea and the United States

As an exploratory investigation, this study aims to analyze non-textual elements in some Korean and US mathematics textbooks using a conceptual framework whose components include accuracy, connectivity, contextuality, and conciseness. By analyzing three US textbooks and three Korean ones, the study not only shows patterns in the use of non-textual elements in mathematics textbooks in different contexts but also provides insights into how to assess the quality of non-textual elements in mathematics textbooks, which I hope will contribute to the provision of more meaningful and productive learning opportunities to school children. Overall, the results from this study show that there is significant difference across topics and textbooks, which implies different opportunities to learn through non-textual elements. This study makes a unique contribution to the conceptualization of non-textual elements in mathematics education and has implications for textbook analysis and curriculum development.     

The Graphing Calculator: A Survey of Classroom Usage

The purpose of this study was to investigate secondary mathematics teachers' use of the graphing calculator in their classrooms, The study examined whether algebra teachers are currently using this technology in their classrooms, their perceptions toward the technology, and any changes in the curriculum or instructional practices. A survey methodology was used in this study. The findings indicated that the use of the graphing calculator is still controversial to many algebra teachers. Teachers of algebra I were using graphing calculators to a significantly lesser degree than teachers of algebra H. However, modifications of the algebra curriculum are beginning to appear in classes using graphing calculators. Finally, a majority of algebra teachers responded that the graphing calculator was a motivational tool.     

Enactment of school mathematics curriculum in Singapore: whither research!

The official curriculum for mathematics in Singapore schools is based on a framework that has mathematical problem solving as its primary goal. It is detailed and one may say that the gap between the designated curriculum and teacher intended curriculum is often very narrow. This is so as the main source of instructional materials is textbooks which are very closely aligned with the official national curriculum. There is a dearth of research on the enactment of the curriculum in Singapore schools, with the few research studies done so far appearing to cover only a narrow focus. The author’s view is that, even though only a few such studies have been published, schools have always been engaged in small-scale investigations, the findings of which are necessary to guide decisions on matters related to choice of textbooks and pedagogies for improved student learning. Considering all the published research and the investigative work undertaken by educators in Singapore, it may be said that the conceptual model proposed by Remillard and Heck is rigorous. In addition, the issues in this particular issue of ZDM offer educators, both classroom teachers and others, very good perspectives for research on the enactment of the school mathematics curriculum.     

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.     

Integrating algebra and proof in high school mathematics: An exploratory study

Frequently, in the US students’ work with proofs is largely concentrated to the domain of high school geometry, thus providing students with a distorted image of what proof entails, which is at odds with the central role that proof plays in mathematics. Despite the centrality of proof in mathematics, there is a lack of studies addressing how to integrate proof into other mathematical domains. In this paper, we discuss a teaching experiment designed to integrate algebra and proof in the high school curriculum. Algebraic proof was envisioned as the vehicle that would provide high school students the opportunity to learn not only about proof in a context other than geometry, but also about aspects of algebra. Results from the experiment indicate that students meaningfully learned about aspects of both algebra and proof in that they produced algebraic proofs involving multiple variables, based on conjectures they themselves generated.     

Mathematics Teachers' Instructional Practices in an Era of High‐Stakes Testing

This study explored mathematics teachers' instructional practices in the context of high‐stakes testing. Data were obtained from a survey instrument given to a stratified sample of Mississippi and Tennessee teachers who teach the same content tested on their state's high school graduation examination. An analysis showed teachers using a balance of standards‐based and traditional practices and tools, including textbook‐based assignments, calculators, open‐response questions, supplementary materials, and multiple‐choice questions. Over 90% of teachers from both states felt that an “interest in helping my students attain test scores that will allow them to graduate high school” was a factor influencing their instructional practices. This was followed by an “interest in helping my school improve high school graduation examination scores,” and “belief these are the best instructional practices.”     

A Comparison of Textbooks' Presentation of Fractions

In the United States, fractions are an important part of the middle school curriculum, yet many middle school students struggle with fraction concepts. Teachers also have difficulty with the conceptual understanding needed to teach fractions and rely on textbooks when making instructional decisions. This reliance on textbooks, the idea that teaching and learning of fractions is a complex process, and that fraction understanding is the foundation for later topics such as proportionality, algebra, and probability, makes it important to examine the variation in presentation of fraction concepts in U.S. textbooks, especially the difference between traditional and standards‐based curricula. The purpose of this study is to determine if differences exist in the presentation of fractions in conventional and standards‐based textbooks and how these differences align with the recommendations of National Council of Teachers of Mathematics, Common Core State Standards, and the research on the teaching and learning of fractions.     

Reasoning-and-Proving in School Mathematics Textbooks

Despite widespread agreement that the activity of reasoning-and-proving should be central to all students' mathematical experiences, many students face serious difficulties with this activity. Mathematics textbooks can play an important role in students' opportunities to engage in reasoning-and-proving: research suggests that many decisions that teachers make about what tasks to implement in their classrooms and when and how to implement them are mediated by the textbooks they use. Yet, little is known about how reasoning-and-proving is promoted in school mathematics textbooks. In this article, I present an analytic/methodological approach for the examination of the opportunities designed in mathematics textbooks for students to engage in reasoning-and-proving. In addition, I exemplify the utility of the approach in an examination of a strategically selected American mathematics textbook series. I use the findings from this examination as a context to discuss issues of textbook design in the domain of reasoning-and-proving that pertain to any textbook series.     

An investigation of teachers’ intentions and reflections about using Standards-based and traditional textbooks in the classroom

This study analyzed teachers’ intentions for and reflections on their use of Standards-based [Connected Mathematics Program (CMP)] textbooks and traditional (non-CMP) mathematics textbooks to guide instruction. In this investigation of the interplay between textbooks and instruction, we focused on learning goals, instructional tasks, teachers’ anticipation of students’ difficulties, and their perceptions of students’ achievement of learning goals. All of these are aspects of teachers’ intentions and reflections that have proved fruitful in comparing the roles of the CMP and non-CMP mathematics textbooks in our Longitudinal Investigation of the Effect of Curriculum on Algebra Learning project. Whereas the cognitive level of the teachers’ intended learning goals appeared generally to reflect the emphases of their respective textbooks, we found that the CMP teachers’ intended learning goals were not as well aligned with the CMP textbooks as the non-CMP teachers’ learning goals were aligned with their non-CMP textbooks. The CMP and non-CMP teachers’ implementations of the lessons seemed to reduce the degree of difference between the cognitive levels of their intended goals. Even so, we found that significantly more CMP lessons than non-CMP lessons were implemented at a high level of cognitive demand. Although the non-CMP teachers’ intended learning goals were better aligned with their textbook’s learning goals, we found that the CMP teachers were more likely than the non-CMP teachers to follow the guidance of their textbooks in designing and selecting instructional tasks for a lesson. Future research should consider other aspects of teachers’ intentions and reflections that may shed a broader light on the role of textbooks and curriculum materials in teachers’ crafting of instructional experiences for their students.     

Relationship Between Professional Development,Teachers' Instructional Practices,and the Achievement of Students in Science and Mathematics

The purpose of this study was to examine the relationship between different types of professional development, teachers' instructional practices, and the achievement of students in science and mathematics. The types of professional development studied included immersion, examining practice, curriculum implementation, curriculum development, and collaborative work. Data regarding teachers' instructional practices and the amount of professional development were collected using teacher surveys. Ninety‐four middle school science teachers and 104 middle school mathematics teachers participated in the study. Student achievement was measured using eighth grade state science and mathematics achievement test data. Regression analyses suggested that for both science and mathematics teachers, examining practice and curriculum development were significantly related to the use of standards‐based instructional practices. Only curriculum development for mathematics teachers was significantly related to student achievement. Implications of results for the professional development of science and mathematics teachers are discussed.     

Ideal and reality: Cross-curriculum work in school mathematics in South Africa

Within various school mathematics dispensations in South Africa the intention for cross-curriculum work is expressed in the official documents describing the intended school mathematics curriculum. This paper traces this expressed intention from 1962 to the present. The view is adopted that textbook authors are the major interpreters of the intended curriculum and therefore the manifestations of the cross-curricular ideal in school textbooks for the various periods are described and commented on. Using the manifestation of the cross-curricular ideal in the South African situation as backdrop, the paper concludes by suggesting ways to deal with three issues that seemingly mitigate against the realisation of this ideal. It is argued that the applications of and modelling in mathematics should be treated as a distinet separate section in the school mathematics curriculum, that mathematics activities, should be designed so that learners with various levels of mathematical sophistication and expertise can deal with both the embedded context and the mathematics and that problems which use context only as a disguise for “pure” mathematics should not be summarily dismissed and written off as useless for the realisation of cross-curricular goals.     

Teaching Mathematics with a New Curriculum: Changes to Classroom Organization and Interactions

This report describes a high school mathematics teacher's decisions about classroom organization and interactions during his first two years using a new curriculum intended to support teachers' development of student-centered, contributive classroom discourse. In year one, the teacher conducted class and interacted with students primarily in small groups. In year two, he conducted more whole-class instruction. In both years, teacher-student interactions contained univocal and contributive discourse, but in year two the teacher sustained contributive discourse with students for longer periods. The teacher facilitated the most significant changes to classroom discourse in the instructional format with which he had the greatest experience (whole-class instruction). Over the period of this study, two key factors appeared to affect the teacher's decisions about classroom organization and interactions: his perception of students' expectations about mathematics classroom roles and activity, and his own discomfort associated with using a new curriculum. These areas are important candidates for future research about teachers' use of innovative mathematics curricula.     

Teaching Algebraic Expressions to Young Students: The Three‐day Journey of “a+ 2”

This classroom scholarship report is based on the teaching experience using Davydov's mathematics curriculum, which was developed in the former Soviet Union. While “from arithmetic to algebra” is the normally accepted instructional sequence in school mathematics, Davydov's curriculum is laid out “from algebra to arithmetic,” focusing on algebraic thinking from the very beginning of the elementary grades. The purpose of this report is not to provide a definitive conclusion about which curriculum or sequence is better nor to address which instructional strategy is right in all circumstances. Rather, it is to explore how primary grade students develop their own conceptual understanding while confronting difficulties met within a specific context. This report provides actual classroom episodes from working with a group of first graders and describes dynamic interactions between the teacher and children while they discuss the use of algebraic expressions and understand the meaning behind them.     

The development of school mathematics textbooks in China since 1950

This paper provides an account of the development of school mathematics textbooks in China since 1950, the year following the founding of the People’s Republic. This development can be divided into several major periods consisting of (a) translating and modifying textbooks from the Soviet Union, (b) writing and editing unified textbooks, and (c) developing multiple versions of textbooks under curriculum standards that emphasize students’ personal development. Over the last 60 years, there have been many changes in the structure and content of developed textbooks; textbooks from each period exhibit their own characteristics which relate to specific political and cultural conditions. The debates on reform of compilation principles and of textbook structure and content still intertwine within the development of school mathematics textbooks. This development has resulted in the launching of a cross-national comparative study on mathematics textbooks in China which is intended to promote the development of mathematics textbooks considering cross-national perspectives.     

The Impact of Enacted Mathematics Curriculum Models on Prospective Elementary Teachers' Course Perceptions and Beliefs

This paper communicates the impact of prospective teachers' learning of mathematics using novel curriculum materials in an innovative classroom setting. Two sections of a mathematics content course for prospective elementary teachers used different text materials and instructional approaches. The primary mathematical authorities were the instructor and text in the textbook section and the prospective teachers in the curriculum materials section. After one semester, teachers in the curriculum materials section (n= 34) placed significantly more importance on classroom group work and discussions, less on instructor lecture and explanation, and less on textbooks having practice problems, examples, and explanations. They valued student exploration over practice. In the textbook section (n= 19), there was little change in the teachers' beliefs, in which practice was valued over exploration. These results highlight the positive impact of experiences with innovative curriculum materials on prospective elementary teachers' beliefs about mathematics instruction.     

Teaching Mathematics with a New Curriculum: Changes to Classroom Organization and Interactions

This report describes a high school mathematics teacher's decisions about classroom organization and interactions during his first two years using a new curriculum intended to support teachers' development of student-centered, contributive classroom discourse. In year one, the teacher conducted class and interacted with students primarily in small groups. In year two, he conducted more whole-class instruction. In both years, teacher-student interactions contained univocal and contributive discourse, but in year two the teacher sustained contributive discourse with students for longer periods. The teacher facilitated the most significant changes to classroom discourse in the instructional format with which he had the greatest experience (whole-class instruction). Over the period of this study, two key factors appeared to affect the teacher's decisions about classroom organization and interactions: his perception of students' expectations about mathematics classroom roles and activity, and his own discomfort associated with using a new curriculum. These areas are important candidates for future research about teachers' use of innovative mathematics curricula.     

Alignment of Elementary Geometry Curriculum With Current Standards

The subject of geometry in the curriculum continues to be an area of concern among mathematics educators. The gap between standards expectations and actual textbook content continues to reflect the need for substantial change. In this study, the geometry language introduced in K‐6 textbooks was carefully examined and compared to the language used in the Curriculum and Evaluation Standards for School Mathematics ( National Council of Teachers of Mathematics, 1989 ), Principles and Standards for School Mathematics ( NCTM, 1999 ), and New Standards Elementary School Mathematics Performance Standards ( National Center for Education and the Economy, 1997 ). The. logic followed was that the language in textbooks reflects the content presented in the textbook, and the language in the standards reflects what should be presented. The results of the study clearly indicate that there is a substantial misalignment between the geometry presented in the textbooks, the. geometry expected to be taught by groups such as NCTM, and the geometry being assessed in student performance measures as suggested by NCEE. Specific areas of misalignment are identified, along with implications, and some areas in which the curriculum should be revised are identified.