## Introduction

Every teacher of mathematics, whether at the elementary, middle, or high school level, has an individual goal to provide students with the knowledge and understanding of the mathematics necessary to function in a world very dependent upon the application of mathematics. Instructionally, this goal translates into three components:

- conceptual understanding
- procedural fluency
- problem solving

Conceptual understanding consists of those relationships constructed internally and connected to already existing ideas. It involves the understanding of mathematical ideas and procedures and includes the knowledge of basic arithmetic facts. Students use conceptual understanding of mathematics when they identify and apply principles, know and apply facts and definitions, and compare and contrast related concepts. Knowledge learned with understanding provides a foundation for remembering or reconstructing mathematical facts and methods, for solving new and unfamiliar problems, and for generating new knowledge.

Procedural fluency is the skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. It includes, but is not limited to, algorithms (the step-by-step routines needed to perform arithmetic operations). Although the word procedural may imply an arithmetic procedure to some, it also refers to being fluent with procedures from other branches of mathematics, such as measuring the size of an angle using a protractor. The use of calculators need not threaten the development of students’ computational skills. On the contrary, calculators can enhance both understanding and computing if used properly and effectively. Accuracy and efficiency with procedures are important, but they should be developed through understanding. When students learn procedures through understanding, they are more likely to remember the procedures and less likely to make common computational errors.

Problem solving is the ability to formulate, represent, and solve mathematical problems. Problems generally fall into three types:

- one-step problems
- multi-step problems
- process problems

Most problems that students will encounter in the real world are multi-step or process problems. Solution of these problems involves the integration of conceptual understanding and procedural knowledge. Students need to have a broad range of strategies upon which to draw. Selection of a strategy for finding the solution to a problem is often the most difficult part of the solution. Therefore, mathematics instruction must include the teaching of many strategies to empower all students to become successful problem solvers. A concept or procedure in itself is not useful in problem solving unless one recognizes when and where to use it as well as when and where it does not apply. Many textbook problems are not typical of those that students will meet in real life. Therefore, students need to be able to have a general understanding of how to analyze a problem and how to choose the most useful strategy for solving the problem.

Individually, each of these components (conceptual understanding, procedural fluency, and problem solving) is necessary but not sufficient for a student to be mathematically proficient. They are not, however, independent of each other. They are integrally related, need to be taught simultaneously, and should be a component of every lesson.

The mathematics standard presented in this document states that students will:

- understand the concepts of and become proficient with the skills of mathematics,
- communicate and reason mathematically;
- become problem solvers by using appropriate tools and strategies;

through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability. Mathematics should be viewed as a whole body of knowledge, not as a set of individual components. Therefore, local mathematics curriculum, instruction, and assessment should be designed to support and sustain the components of this standard.

New York State’s yearly 3-8 mathematics assessments, as required by NCLB federal legislation, will provide data measuring student progress toward obtaining mathematical proficiency. Since the state assessments will measure conceptual understanding, procedural fluency, and problem solving, local assessments should measure these components as well. Thus, many schools may need to provide teachers with significant professional staff development to assist them in developing local assessments.

In this document conceptual understanding, procedural fluency, and problem solving are represented as process strands and content strands. These strands help to define what students should know and be able to do as a result of their engagement in the study of mathematics.

Process Strands: The process strands (Problem Solving, Reasoning and Proof, Communication, Connections, and Representation) highlight ways of acquiring and using content knowledge. These process strands help to give meaning to mathematics and help students to see mathematics as a discipline rather than a set of isolated skills. Student engagement in mathematical content is accomplished through these process strands. Students will gain a better understanding of mathematics and have longer retention of mathematical knowledge as they solve problems, reason mathematically, prove mathematical relationships, participate in mathematical discourse, make mathematical connections, and model and represent mathematical ideas in a variety of ways.

Content Strands: The content strands (Number Sense and Operations, Algebra, Geometry, Measurement, and Statistics and Probability) explicitly describe the content that students should learn. Each school’s mathematics curriculum developed from these strands should include a broad range of content. This broad range of content, taught in an integrated fashion, allows students to see how various mathematics knowledge is related, not only within mathematics, but also to other disciplines and the real world as well. The performance indicators listed under each band within a strand are intended to assist teachers in determining what the outcomes of instruction should be. The instruction should engage students in the construction of this knowledge and should integrate conceptual understanding and problem solving with these performance indicators. The performance indicators should not be viewed as a checklist of skills void of understanding and application.

Students will only become successful in mathematics if they see mathematics as a whole, not as isolated skills and facts. As school districts develop their own mathematics curriculum based upon the statements in this standards document, attention must be given to both content and process strands. Likewise, as teachers develop their instructional plans and their assessment techniques, they also must give attention to the integration of process and content. To do otherwise would produce students who have temporary knowledge and who are unable to apply mathematics in realistic settings. Curriculum, instruction, and assessment are intricately related and must be designed with this in mind. All three domains must address conceptual understanding, procedural fluency, and problem solving. If this is accomplished, school districts will produce students who will (1) have mathematical knowledge, (2) have an understanding of mathematical concepts, and (3) be able to apply mathematics in the solution of problems.

School districts and individual teachers should be aware that this document is a standards document that guides the development of local curriculum. Local school districts remain responsible for developing curriculum aligned to the New York State standards. In this document the mathematics standard is succinctly stated. The standard outlines what students should know and be able to do in mathematics. The content strands, consisting of bands and performance indicators within each band, and the performance indicators of the process strands help to define how the standard will be met. Each school district’s mathematics curriculum should be developed to assure that all students achieve the performance indicators for both the process and content strands.

Helping all students become proficient in mathematics is an imperative goal for every school. It is the hope that this standards document will assist schools and individual teachers in meeting this goal. For additional information visit the New York State Education Department mathematics website.

## Proposed Mathematics Standard, Content Strands, Process Strands, Bands within the Content Strands, and Grade-By-Grade Performance Indicators

### Mathematics, Science, and Technology - Standard 3

Students will:

- understand the concepts of and become proficient with the skills of mathematics;
- communicate and reason mathematically;
- become problem solvers by using appropriate tools and strategies;

through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.

### The Five Content Strands

#### Number Sense and Operations Strand

Students will:

- understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems;
- understand meanings of operations and procedures, and how they relate to one another;
- compute accurately and make reasonable estimates.

#### Algebra Strand

Students will:

- represent and analyze algebraically a wide variety of problem solving situations;
- perform algebraic procedures accurately;
- recognize, use, and represent algebraically patterns, relations, and functions.

#### Geometry Strand

Students will:

- use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes;
- identify and justify geometric relationships, formally and informally;
- apply transformations and symmetry to analyze problem solving situations;
- apply coordinate geometry to analyze problem solving situations.

#### Measurement Strand

Students will:

- determine what can be measured and how, using appropriate methods and formulas;
- use units to give meaning to measurements;
- understand that all measurement contains error and be able to determine its significance;
- develop strategies for estimating measurements.

#### Statistics and Probability Strand

Students will:

- collect, organize, display, and analyze data;
- make predictions that are based upon data analysis;
- understand and apply concepts of probability.

### The Five Process Strands

#### Problem Solving Strand

Students will:

- build new mathematical knowledge through problem solving;
- solve problems that arise in mathematics and in other contexts;
- apply and adapt a variety of appropriate strategies to solve problems;
- monitor and reflect on the process of mathematical problem solving.

#### Reasoning and Proof Strand

Students will:

- recognize reasoning and proof as fundamental aspects of mathematics;
- make and investigate mathematical conjectures;
- develop and evaluate mathematical arguments and proofs;
- select and use various types of reasoning and methods of proof.

#### Communication Strand

Students will:

- organize and consolidate their mathematical thinking through communication;
- communicate their mathematical thinking coherently and clearly to peers, teachers, and others;
- analyze and evaluate the mathematical thinking and strategies of others;
- use the language of mathematics to express mathematical ideas precisely.

#### Connections Strand

Students will:

- recognize and use connections among mathematical ideas;
- understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
- recognize and apply mathematics in contexts outside of mathematics.

#### Representation Strand

Students will:

- create and use representations to organize, record, and communicate mathematical ideas;
- select, apply, and translate among mathematical representations to solve problems;
- use representations to model and interpret physical, social, and mathematical phenomena.

### Bands Within the Content Strands

#### Number Sense and Operations

- Number Systems
- Number Theory
- Operations
- Estimation

#### Algebra

- Variables and Expressions
- Equations and Inequalities
- Patterns, Relations, and Functions
- Coordinate Geometry
- Trigonometric Functions

#### Geometry

- Shapes
- Geometric Relationships
- Transformational Geometry
- Coordinate Geometry
- Constructions
- Locus
- Informal Proofs
- Formal Proofs

#### Measurement

- Units of Measurement
- Tools and Methods
- Units
- Error and Magnitude
- Estimation

#### Statistics and Probability

- Collection of Data
- Organization and Display of Data
- Analysis of Data
- Predictions from Data
- Probability