BPS Learning Standards and Curriculum Framework: Mathematics Introduction to Mathematics Curriculum and Standards Focus: to create consistency and continuity across the grade levels, to set parameters for the designation of instructional time, and most importantly, to improve the mathematics education of all students in the Boston Public Schools.  Guiding Principles: There are underlying beliefs and tenets that are central to the vision of mathematical power and the content standards for mathematics education: Students should explore mathematical ideas in ways that maintain their enjoyment of and curiosity about mathematics, help them develop depth of understanding, and reflect real-world applications. All students should have access to a high quality mathematics program. Mathematics learning is a lifelong process that begins and continues in the home and extends to the school and community settings. Mathematics instruction should make connections with other disciplines, within mathematics, and to real-world situations. Working together in teams and groups enhances mathematical learning, helps students communicate effectively, and develops social and mathematical skills. Technology is an essential tool for effective mathematics education. Mathematics assessment is a multifaceted tool that monitors student performance, improves instruction, enhances learning, and encourages student self-reflection.   1996 Massachusetts Mathematics Curriculum Framework Core Processes: The Boston Public Schools Citywide Learning Standards for Mathematics build upon the Massachusetts Mathematics Curriculum Framework's core concept that students develop mathematical power through problem solving, communication, reasoning, and connections. These concepts are elaborated on further at the beginning of the PreK-5 and 6-9 Learning Standards. Problem Solving: Problem solving is the central focus of mathematics education. Whenever we apply our mathematics knowledge, skills, or experiences to the resolution of a dilemma or situation that is new or perplexing, we are problem solving. Communication: Students learn mathematics best when they talk and write about what they are doing. Communication in mathematics promotes mathematical understanding and creates mathematical power. Reasoning: Mathematics reasoning is necessary if we are to know and do mathematics. The ability to reason enables students to solve problems in their lives, in and outside of school. Students need to be provided with opportunities to make conjectures, think about and select sensible ways to solve problems, justify their solutions, and share their reasoning with others. They also need to experience, recognize, learn, and use different types of reasoning. Connections: Mathematics topics are connected to real life and to one other. Students should understand how mathematics relates to other subject areas. For example, all students should understand the concepts of location and place, each of which is important to both geography and mathematics. Students should appreciate that mathematics is one way of learning about the world, and that it is connected, not isolated, from other ways of learning. Students need to see that when examined, many problems in the world initially appear quite different; but when we model these same problems mathematically, they have striking similarities. Organizational Structure: Our Standards are organized by "strands" that are included in the Massachusetts Curriculum Frameworks. Number Sense Patterns, Relations, and Functions Geometry, Spatial Sense, and Measurement Probability, Statistics, and Discrete Mathematics This organization promotes an "integrated" approach to mathematics. Integrated Mathematics: An integrated approach to teaching mathematics means making connections within mathematics, as well as to other disciplines and real world situations. In an "integrated mathematics" program, students are taught increasingly sophisticated concepts from different strands of mathematics, including arithmetic, algebra, geometry, probability, and statistics. Measurement is related to number, data, and geometry; and algebraic ideas of patterns and relationships are developed in all areas of mathematics. This process begins in the early grades and continues throughout high school.  Students learn to solve real-life problems that require the integration of skills from two or more of these mathematics strands. Skills, concepts, and procedures are taught through authentic tasks as an integrated whole, not in isolation of one another. Concepts are taught over a long period of time, and connected to a variety of strands. For example, the concept of fractions should be explored and taught a number of times throughout the school year:  as parts-to-whole (geometry) as the comparison of two quantities (number sense, ratio) as the comparison of a particular quantity with a fixed quantity (decimals and percent) as the comparison of two comparisons (ratio and proportion) through the application of fractions as rates one can investigate (probability and statistics) When students are exposed to and use fractions in the many ways they are commonly used in the real world, they can become far more proficient problem-solvers. Beginning in 1998, students throughout Massachusetts will take statewide assessments in mathematics. These assessments will reflect the integrated approach adopted by the state. We are aligning our Standards with this approach to prepare our students for these assessments. Finally, the integrated approach is not new. In most European countries, mathematics is taught in an integrated manner. For example, there is no artificial separation of algebra and geometry. By the same token, math instruction in many of our classrooms is already integrate. Effective teachers, particularly at the elementary and middle school levels, have been using an integrated approach to mathematics instruction for years. For example, even very young children begin to study shapes and sizes during the primary grades, though the language of geometry may not be used. Many Boston middle schools already introduce algebra concepts through an integrated approach. All of this work will continue, and be supported by an integrated approach across all levels.   Key Characteristics  The following principles define our Standards' key characteristics. Accessible: Our Standards are accessible to the children in the Boston Public Schools. Rigorous: Our Standards are challenging. In addition to requiring the mastery of facts and procedures, we expect students to know the meaning of computation and operations. Students are required to move beyond the intuitive and concrete level of understanding, to the representational, abstract, application, and communication levels. Mastery is achieved only when a student is able to communicate his or her learning. Meaningful: Our Standards require students to make meaningful connections between their classroom and the world outside of school. The Standards demonstrate to students that learning mathematics is a life-long process, and its purpose is to solve real problems. Inquiry-Centered: Our Standards promote experiences that are exploratory and discovery-oriented. Students should be encouraged to persist and take risks, be actively engaged, and feel confident in their ability to do mathematics. Though students will make many mistakes along the way, we view these mistakes as opportunities to learn, particularly when they are discussed with other students and the teacher. Communication and Reflection-Centered: Mathematics is a language to be learned, with its own vocabulary, syntax, and rules of translation from language to symbol (and vice-versa). We expect students to explore and reflect on concepts and procedures, to express their mathematics understandings clearly, through speech and writing, and to develop their reading skills, as related to mathematics. Student Focused/Appropriately Paced: Some students need more time to master the concepts included in our Standards. Our Standards framework responds to these needs. Every student will graduate with a solid foundation in arithmetic, algebra, geometry, probability, and statistics. Some students will also graduate with a solid foundation in calculus. Concrete: Our Standards provide teachers and students with concrete example of problems, tasks, and products students will be expected to solve and complete. Assessment-Embedded: Assessment is an integral part of the mathematics program, and is aligned with curriculum and instruction. A variety of methods such as performance tasks, projects, observations, discussions, writings, etc. are used to measure and monitor student performance, enhance learning, and improve instruction. Mathematics assessment also encourages student reflection. Students are expected to take increasing responsibility for organizing and extending tasks into projects and long-term activities. They will select and organize mathematics and resources, extend work to related tasks, review their progress, and check and evaluate their work. Throughout this process, students will develop confidence, perseverance, and persistence by continuing to work on problems that pose challenges. Technological: Our Standards require students to understand and use appropriate instructional technology, including calculators, computer software, graphing utilities, statistics packages, and different types of measuring devices, to enhance and accelerate mathematics learning, to represent and communicate mathematical ideas, and to make mathematics more accessible to all children. Appropriate technologies will be used in the spirit of manipulatives to make presentations, carry out explorations, and assess mathematical concepts and skills. Open-Ended: As children need more and more mathematics to be successful in life, as we learn more about the way children learn mathematics, and as instructional methodologies get better, our Standards and curriculum need to keep pace with these changes. Our Standards are designed to be implemented, reviewed, and revised, as needed, on an ongoing three to five year cycle.     Core Processes Problem Solving: Problem solving is not a distinct topic, but a process that should permeate the entire program and provide the context in which concepts and skills can be learned. (NCTM) Students will... form and solve problems using multiple approaches work with math manipulatives in a variety of ways verify and interpret results work out ways to overcome their difficulties organize and check their work search for patterns while working with numbers, data, shapes, and objects in their environment sort, classify, and make comparisons use calculators and computers to generate and explain number apply mathematics to practical tasks and real-life problems gain confidence in solving math problems   Communication: All students should develop and present conclusions through speaking, writing, artistic, and other means of expression. (Massachusetts Common Core of Learning) Students will... express mathematical ideas, skills, and discoveries through demonstration, drawing, writing, and talking relate everyday language to mathematical language and symbols work together to create and solve problems exchange problems with each other justify their own reasoning about mathematics record and present their work in a variety of ways discuss their work, responding to and asking critical questions such as: What would happen if...? Why? What is the pattern? Can this be solved or expressed another way?   Reasoning: If we would guide by the light of reason, we must let our minds be bold. (Louis Brandeis) Students will... make and evaluate mathematical conjectures and arguments think about and select sensible ways to solve problems justify solutions and explain the problem solving process share their reasoning with others experience, recognize, learn, and use different types of reasoning Connections: Students who are able to apply and translate among different representations of the same problem situation, or the same mathematical concept, will have at once a powerful, flexible set of tools for solving problems, and a deeper appreciation of the consistency and beauty of mathematics. (NCTM) Students will... use mathematics skills in everyday life recognize the relationships among different topics in mathematics connect ideas within the curriculum strand and with other concepts they are learning work with appropriate technology in the development of concepts and skills     Agassiz School Mathematics Plan Implementation of BPS Mathematics Plan: Agassiz School     Overview The Boston Public Schools Mathematics Plan has been designed to support high quality rigorous mathematics education for all students. As a school which has already successfully implemented a Literacy Plan, the Agassiz has begun it's second stage in implementing the BPS Mathematics Plan.   Math Leadership Team The BPS Mathematics Plan supports the development of organizational structures in schools that support and sustain the collaboration of students, teachers, administrators and parents around mathematics teaching and learning. A Math Leadership Team has been established consisting of the Math Specialist, six classroom teachers (one from each grade level K-5) the Principal and a Math Coach who will mentor and support the Math Leadership Team as they receive professional development and begin to implement the"Investigations in Number, Data and Space" materials developed by TERC in their classroom..   Implementing the "Investigations in Number, Data and Space" Curriculum The Mathematics Plan calls for the implementation of TERC's "Investigations in Number, Data and Space" in all classrooms the by the 2001 school year. In the 2000 school year, one classroom teacher in each grade level will implement approximately four units of Investigations with their students in conjunction with teacher training, guidance and professional development.   TERC Founded in 1965, TERC is a not-for-profit education research and development organization in Cambridge, Massachusetts. TERC's mission is to improve mathematics, science, and technology teaching and learning. TERC works at the edges of current theory and practice to: contribute to understanding of learning and teaching foster professional development develop applications of new technologies create curricula and other products support school reform "We imagine a future in which learners from diverse communities engage in creative, rigorous, and reflective inquiry as an integral part of their lives." A Mission Centered on Learning Learning is at the heart of TERC's mission. We view learning as an ongoing way of being in the world that inspires us to focus on the present, reflect on our past, and imagine future possibilities. At TERC our work in mathematics and science education aims to instill in all students a desire for life-long learning. Through research, professional development, curriculum development, and applications of technology, TERC provides opportunities for learners of all ages to come together and engage in rigorous and reflective inquiry. Environments that promote inquiry and collaboration allow learners to assume a research perspective. That is, through their investigations, observations, and analysis learners construct their own scientific and mathematical understandings of how the world works. Within the community the learner develops a way to communicate those understandings, contributing to both the individual and collective knowledge bases.   K-5 Math Curriculum: Investigations in Number, Data, and Space Investigations in Number, Data, and Space is a comprehensive, K-5 mathematics curriculum whose goals are to: offer students meaningful mathematical problems; emphasize depth in mathematical thinking rather than exposure to a series of fragmented topics; communicate mathematics content and pedagogy to teachers; and widely expand the pool of mathematically literate students. The curriculum consists of a set of modules at each grade level. Each module offers a series of connected investigations of major mathematical ideas within the areas of number, data collection and analysis, geometry, and the mathematics of change. These investigations offer significant content and encourage students to develop flexibility and confidence in approaching mathematical problems, proficiency in evaluating solutions, and a repertoire of ways to communicate about their mathematical thinking. Funded by the National Science Foundation. Project Director: Susan Jo Russell Investigations in Number, Data, and Space is part of TERC's Education Research Collaborative in Science and Mathematics.   Professional Development As part of this approach to teaching and learning, teachers and administrators will receive professional development designed to support implementation of the Investigations curriculum, monitoring of student learning and informing instruction. This year Agassiz teachers and administrators will also receive professional development training through the Developing Mathematical Ideas seminars created by CDT (Center for the Development of Teaching) a division of EDC (Educational Development Center, Inc).   Developing Mathematical Ideas: A Curriculum for Teacher Learning Developing Mathematical Ideas (DMI) is a curriculum designed to help teachers think through the major ideas of K-6 mathematics and examine how children develop those ideas. At the heart of the materials are sets of classroom episodes (cases), illustrating student thinking as described by their teachers. In addition to case discussions, the curriculum offers teachers opportunities: to explore mathematics in lessons led by facilitators; to share and discuss the work of their own students; to plan, conduct, and analyze mathematics interviews of their own students; to view and discuss videotapes of mathematics classrooms and mathematics interviews; to write their own classroom episodes; to analyze lessons taken from innovative elementary mathematics curricula; and to read overviews of related research. The major goals of the DMI seminars are to help participants: earn more mathematics content learn to define and select mathematical objectives for their students learn to recognize key mathematical ideas with which their students are grappling learn how to support children's mathematical thinking learn to appreciate the power and complexity of student thinking learn how to ask questions that will help students deepen their mathematical understanding learn how to analyze a piece of curriculum for the mathematics students will learn from it learn to make more mathematical connections for themselves, enhancing their ability to help their students do so learn how to continue learning about children and mathematics   Developing Mathematical Ideas Seminars DMI materials for two seminars have been completed- Number and Operations, Part 1: Building a System of Tens and Number and Operations, Part 2: Making Meaning for Operations. Under production are materials that will address geometry, measurement, and data analysis--Examining Features of Shape, Measuring Space in One, Two and Three Dimensions, and Working with Data. Number and Operations, Part 1: Building a System of Tens In this seminar, participants explore the base-ten structure of the number system, consider how that structure is exploited in multidigit computational procedures, and examine how basic concepts of whole numbers reappear when working with decimals. Number and Operations, Part 2: Making Meaning for Operations In this seminar, participants examine the actions and situations modeled by the four basic operations. The seminar begins with a view of young children's counting strategies as they encounter word problems, moves to an examination of children's developing ideas of the four basic operations, and revisits the operations in the context of rational numbers. Examining Features of Shape In this seminar, participants examine aspects of 2D and 3D shapes, develop geometric vocabulary, and explore both definitions and properties of geometric objects. The seminar includes a study of angle, similarity, congruence and the relationships between 3D objects and their 2D representations. Measuring Space in One, Two and Three Dimensions In this seminar, participants examine different attributes of size, develop facility in composing and decomposing shapes, and apply these skills to make sense of formulas for area and volume. They also explore conceptual issues of length, area, volume, as well as their complex inter-relationships. Working with Data In this seminar, participants work with the collection, description, and interpretation of data. They learn what various graphical representations and statistical measures show about features of the data, especially when comparing groups.     MASSACHUSETTS STATE LEARNING STANDARDS: Mathematics   Guiding Philosophy This curriculum framework envisions all students in the Commonwealth achieving mathematical competence through a strong mathematics program that emphasizes problem solving, communicating, reasoning and proof, making connections, and using representations. Acquiring such competence depends in large part on a clear, comprehensive, coherent, and developmentally appropriate set of standards to guide curriculum expectations. Problem Solving Problem solving is both a means of developing students' knowledge of mathematics and a critical outcome of a good mathematics education. As such, it is an essential component of the curriculum. A mathematical problem, as distinct from an exercise, requires the solver to search for a method for solving the problem rather than following a set procedure. Mathematical problem solving, therefore, requires an understanding of relevant concepts, procedures, and strategies. To become good problem solvers, students need many opportunities to formulate questions, model problem situations in a variety of ways, generalize mathematical relationships, and solve problems in both mathematical and everyday contexts. Communicating The ability to express mathematical ideas coherently to different audiences is an important skill in a technological society. Students develop this skill and deepen their understanding of mathematics when they use accurate mathematical language to talk and write about what they are doing. They clarify mathematical ideas as they discuss them with peers, and reflect on strategies and solutions. By talking and writing about mathematics, students learn how to make convincing arguments and to represent mathematical ideas verbally, pictorially, and symbolically. Reasoning and Proof From the early grades on, students develop their reasoning skills by making and testing mathematical conjectures, drawing logical conclusions, and justifying their thinking in developmentally appropriate ways. As they advance through the grades, students' arguments become more sophisticated and they are able to construct formal proofs. By doing so, students learn what mathematical reasoning entails. Making Connections Mathematics is not a collection of separate strands or standards. Rather, it is an integrated field of study. Students develop a perspective of the mathematics field as an integrated whole by understanding connections within and outside of the discipline. It is important for teachers to demonstrate the significance and relevance of the subject by encouraging students to explore the connections that exist within mathematics, with other disciplines, and between mathematics and students' own experiences. Representations Mathematics involves using various types of representations for mathematical objects and actions, including numbers, shapes, operations, and relations. These representations can be numerals or diagrams, algebraic expressions or graphs, or matrices that model a method for solving a system of equations. Students must learn to use a repertoire of mathematical representations. When they can do so, they have a set of tools that significantly expands their capacity to think mathematically.   Guiding Principles Guiding Principle I: Learning Mathematical ideas should be explored in ways that stimulate curiosity, create enjoyment of mathematics, and develop depth of understanding. Students need to understand mathematics deeply and use it effectively. To achieve mathematical understanding, students should be actively engaged in doing meaningful mathematics, discussing mathematical ideas, and applying mathematics in interesting, thought provoking situations. Student understanding is further developed through ongoing reflection about cognitively demanding and worthwhile tasks. Tasks should be designed to challenge students in multiple ways. Short- and long-term investigations that connect procedures and skills with conceptual understanding are integral components of an effective mathematics program. Activities should build upon curiosity and prior knowledge, and enable students to solve progressively deeper, broader, and more sophisticated problems. Mathematical tasks reflecting sound and significant mathematics should generate active classroom talk, promote the development of conjectures, and lead to an understanding of the necessity for mathematical reasoning. Guiding Principle II: Teaching An effective mathematics program focuses on problem solving and requires teachers who have a deep knowledge of mathematics as a discipline. Mathematical problem solving is the hallmark of an effective mathematics program. Skill in mathematical problem solving requires practice with a variety of mathematical problems as well as a firm grasp of mathematical techniques and their underlying principles. Armed with this deeper knowledge, the student can then use mathematics in a flexible way to attack various problems and devise different ways of solving any particular problem. Mathematical problem solving calls for reflective thinking, persistence, learning from the ideas of others, and going back over one's own work with a critical eye. Success in solving mathematical problems helps to create an abiding interest in mathematics. For a mathematics program to be effective, it must also be taught by knowledgeable teachers. According to Liping Ma, "The real mathematical thinking going on in a classroom, in fact, depends heavily on the teacher's understanding of mathematics."3 A landmark study in 1996 found that students with initially comparable academic achievement levels had vastly different academic outcomes when teachers' knowledge of the subject matter differed.4 The message from the research is clear: having knowledgeable teachers really does matter; teacher expertise in a subject drives student achievement. National data show that "nearly one-third of all secondary school teachers who teach mathematics have neither a major nor a minor in the subject itself, in mathematics education, or even in a related discipline."5 While there are very effective teachers who do not have a major or minor in mathematics or in a related field, the goal should be that all future teachers have concentrated study in the field of mathematics. "Improving teachers' content subject matter knowledge and improving Mathematics Curriculum Framework students' mathematics education are thus interwoven and interdependent processes that must occur simultaneously." Guiding Principle III: Technology Technology is an essential tool in a mathematics education. Technology enhances the mathematics curriculum in many ways. Tools such as measuring instruments, manipulatives (such as base ten blocks and fraction pieces), scientific and graphing calculators, and computers with appropriate software, if properly used, contribute to a rich learning environment for developing and applying mathematical concepts. However, appropriate use of calculators is essential; calculators should not be used as a replacement for basic understanding and skills. Moreover, the fourth and sixth grade state assessments do not permit the use of a calculator. Elementary students should learn how to perform thoroughly the basic arithmetic operations independent of the use of a calculator.7 Although the use of a graphing calculator can help middle and secondary students to visualize properties of functions and their graphs, graphing calculators should be used to enhance their understanding and skills rather than replace them. Technology enables students to communicate ideas within the classroom or to search for information in external databases such as the Internet, an important supplement to a school's internal library resources. Technology can be especially helpful in assisting students with special needs in regular and special classrooms, at home, and in the community. Technology changes what mathematics is to be learned and when and how it is learned. For example, currently available technology provides a dynamic approach to such mathematical concepts as functions, rates of change, geometry, and averages that was not possible in the past. Some mathematics becomes more important because technology requires it, some becomes less important because technology replaces it, and some becomes possible because technology allows it. Guiding Principle IV: Equity All students should have a high quality mathematics program. All Massachusetts students should have high quality mathematics programs that meet the goals and expectations of these standards and address students' individual interests and talents. The standards provide for a broad range of students, from those requiring tutorial support to those with talent in mathematics. To promote achievement of these standards, teachers should encourage classroom talk, reflection, use of multiple problem solving strategies, and a positive disposition toward mathematics. They should have high expectations for all students. At every level of the education system, teachers should act on the belief that every child should learn challenging mathematics. Teachers and guidance personnel should advise students and parents about why it is important to take advanced courses in mathematics and how this will prepare students for success in college and the workplace. All students should have the benefit of quality instructional materials, good libraries, and adequate technology. Practice and enrichment should extend beyond the classroom. Tutorial sessions, mathematics clubs, competitions, and apprenticeships are examples of mathematics activities that promote learning. Because mathematics is the cornerstone of many disciplines, a comprehensive curriculum should include applications to everyday life and modeling activities that demonstrate the connections among disciplines. Schools should also provide opportunities for communicating with experts in applied fields to enhance students' knowledge of these connections. Guiding Principle V: Assessment Assessment of student learning in mathematics should take many forms to inform instruction and learning. A comprehensive assessment program is an integral component of an instructional program. It provides students with frequent feedback on their performance, teachers with diagnostic tools for gauging students' depth of understanding of mathematical concepts and skills, parents with information about their children's performance in the context of program goals, and administrators with a means for measuring student achievement. Assessments take a variety of forms, require varying amounts of time, and address different aspects of student learning. Having students "think aloud" or talk through their solutions to problems permits identification of gaps in knowledge and errors in reasoning. By observing students as they work, teachers can gain insight into students' abilities to apply appropriate mathematical concepts and skills, make conjectures, and draw conclusions. Homework, mathematics journals, portfolios, oral performances, and group projects offer additional means for capturing students' thinking, knowledge of mathematics, facility with the language of mathematics, and ability to communicate what they know to others. Tests and quizzes assess knowledge of mathematical facts, operations, concepts, and skills and their efficient application to problem solving. They can also pinpoint areas in need of more practice or teaching. Taken together, the results of these different forms of assessment provide rich profiles of students' achievements in mathematics and serve as the basis for identifying curricula and instructional approaches to best develop their talents. Assessment should also be a major component of the learning process. As students help identify goals for lessons or investigations, they gain greater awareness of what they need to learn and how they will demonstrate that learning. Engaging students in this kind of goal-setting can help them reflect on their own work, understand the standards to which they are held accountable, and take ownership of their learning.     Boston Public Schools Citywide Learning Standards and Curriculum Guidelines The BPS Curriculum Guides in Mathematics (Grades K-5) are available for reading online at the Boston Public Schools Website. To read them, press the chalkboard icon below. Use the "back" button of your browser to return to this site.      • BPS Curriculum Guide & Citywide Learning Standards: Mathematics Grade 5   •BPS Curriculum Guide & Citywide Learning Standards: Mathematics Grade 4   •BPS Curriculum Guide & Citywide Learning Standards: Mathematics 3   • BPS Curriculum Guide & Citywide Learning Standards: Mathematics Grade 2   •BPS Curriculum Guide & Citywide Learning Standards: Mathematics Grade 1   •BPS Curriculum Guide & Citywide Learning Standards: Mathematics Grade K