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The Data Cycle
The Data Cycle
Analyzing data requires the ability to read, write, and communicate about data in context. The skills needed to analyze data are integrated in the mathematics standards and derived from and build upon a strong mathematical foundation. A process for data analysis is included in the standards as a Data Cycle. The cycle includes asking meaningful questions, collecting, or acquiring appropriate data, and analyzing, interpreting, and communicating about the data. The standards include direct reference to mathematical skills needed to analyze and interpret data. Data and data analysis are necessary for many jobs such as those in science, technology, business, and engineering, but also to ensure students Efficiency Flexibility Accuracy Virginia Department of Education (VDOE) – Mathematics Standards of Learning – August 2023 Page 112 can develop problem solving skills and navigate as a citizen in a world in which data plays a vital role.
Mathematical Process Goals
Mathematical Process Goals
Students learn and apply the five mathematical process goals as they work to achieve the content of the Mathematics Standards. These processes support students in becoming mathematical problem solvers, communicating mathematically, reasoning mathematically, making mathematical connections, and using mathematical representations to model and interpret contextual situations. Contextual situations include real-world problems and problems that model real-world situations.
Mathematical Problem Solving
Students will apply mathematical concepts and skills and the relationships among them to solve problem situations of varying complexities. Students also will recognize and create problems from real-world data and situations within and outside mathematics and then apply appropriate strategies to determine acceptable solutions. To accomplish this goal, students will need to develop a repertoire of skills and strategies for solving a variety of problems. A major goal of the mathematics program is to help students apply mathematics concepts and skills to become mathematical problem solvers.
Mathematical Communication
Students will communicate thinking and reasoning using the language of mathematics, including specialized vocabulary and symbolic notation, to express mathematical ideas with precision. Representing, discussing, justifying, conjecturing, reading, writing, presenting, and listening to mathematics will help students clarify their thinking and deepen their understanding of the mathematics being studied. Mathematical communication becomes visible when learning involves participation in mathematical discussions.
Mathematical Reasoning
Students will recognize reasoning and proof as fundamental aspects of mathematics. Students will learn and apply inductive and deductive reasoning skills to make, test, and evaluate mathematical statements and to justify steps in mathematical procedures. Students will use logical reasoning to analyze an argument and to determine whether conclusions are valid. In addition, students will use number sense to apply proportional and spatial reasoning and to reason from a variety of representations.
Mathematical Connections
Students will build upon prior knowledge to relate concepts and procedures from different topics within mathematics and see mathematics as an integrated field of study. Through the application of content and process skills, students will make connections among different areas of mathematics and between mathematics and other disciplines, and to real-world contexts. Science and mathematics teachers and curriculum writers are encouraged to develop mathematics and science curricula that support, apply, and reinforce each other.
Mathematical Representations
Students will represent and describe mathematical ideas, generalizations, and relationships using a variety of methods. Students will understand that representations of mathematical ideas are an essential part of learning, doing, and communicating mathematics. Students should make connections among different representations – physical, visual, symbolic, verbal, and contextual – and recognize that representation is both a process and a product.
Instructional Technology
Instructional Technology
The strategic use of technological tools can support both the learning of mathematical procedures and skills as well as the development of advanced mathematical proficiencies, such as problem solving, reasoning, and justifying.
State and local assessments may restrict the use of calculators in measuring specific student objectives that focus on number sense and computation. On the grade three state assessment, all objectives are assessed without the use of a calculator. On the state assessments for grades four through seven, the Knowledge and Skills that are assessed without the use of a calculator are indicated with an asterisk (*).
Computational Fluency
Computational Fluency
The mathematics standards emphasize procedural understanding through the development of computational fluency. Computational fluency is the ability to use flexible, efficient, and accurate methods for computing. Students exhibit computational fluency when they demonstrate strategic thinking and flexibility in the computational methods they choose and apply and can explain why their actions make sense. Students demonstrate effective use of strategies and methods while reflecting on which procedures seem to work best for specific types of problems. Students share their ideas collaboratively, discuss how to solve problems in different ways and produce accurate answers efficiently.
The computational methods used by a student should be based on the mathematical ideas that the student understands, including the structure of the base-ten number system, number relationships, meaning of operations, and properties. Computational fluency with whole numbers is a goal of mathematics instruction in the elementary grades.
Rigorous mathematics instruction must simultaneously develop students’ conceptual understanding, computational fluency, and problem-solving skills. The development of related conceptual understanding and computational skills should be balanced and intertwined, each supporting the other and reinforcing learning.
Algebraic Thinking & Readiness
Algebraic Thinking & Readiness
The study of algebraic thinking begins in kindergarten and is progressively formalized prior to the study of the algebraic content found in the Algebra 1 Standards of Learning. The preparation of students for Algebra 1 includes the mastery of, and the ability to apply, the Mathematics Standards of Learning, including the Mathematical Process Goals for Students, for kindergarten through grade eight. Included in the progression of algebraic content in kindergarten through grade eight is patterning; generalization of arithmetic concepts; proportional reasoning; representing mathematical relationships using tables, symbols, and graphs; and algebraic equations and inequalities. The K-8 Mathematics Standards of Learning form a progression of content knowledge and develop the foundation and reasoning necessary to be well-prepared for mathematics courses beyond Algebra 1, including Geometry and Statistics. Divisions have local autonomy to compact content in the mathematics standards of learning when creating accelerated curriculum for students taking Algebra 1 in middle school. Algebra 1 is a course that all students should have the opportunity to take in Grade 8 or earlier. School divisions should ensure processes are in place to create accelerated coursework and to support individual readiness.
Formal Mathematics Vocabulary
Formal Mathematics Vocabulary
The development of mathematics vocabulary supports students’ conceptual understanding, abstract reasoning, and ability to communicate effectively. Teachers should facilitate student connections between formal mathematical vocabulary and their current conceptual understanding. Teacher and student use of formal mathematics vocabulary is a focus of instruction in all mathematics classrooms.
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