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STANDARD 1: MATHEMATICS AS PROBLEM SOLVING |
Examples of Problem Solving |

In grades K-4, the study of mathematics should emphasize problem solving so that students can--

use problem-solving approaches to investigate and understand mathematical content;formulate problems from everyday and mathematical situations;develop and apply strategies to solve a wide variety of problems;verify and interpret results with respect to the original problem;acquire confidence in using mathematics meaningfully.

Problem solving should be the central focus of the mathematics curriculum. As such, it is a primary goal of all mathematics instruction and an integral part of all mathematical activity. 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.

This standard emphasizes a comprehensive and rich approach to problem solving in a classroom climate that encourages and supports problem-solving efforts. Ideally, students should share their thinking and approaches with other students and with teachers, and they should learn several ways of representing problems and strategies for solving them. In addition, they should learn to value the process of solving problems as much as they value the solutions. Students should have many experiences in creating problems from real-world activities, from organized data, and from equations.

STANDARD 2: MATHEMATICS AS COMMUNICATION |
Examples of Mathematics as Communication |

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In grades K-4, the study of mathematics should include numerous opportunities for communication so that students can--

relate physical materials, pictures, and diagrams to mathematical ideas;reflect on and clarify their thinking about mathematical ideas and situations;relate their everyday language to mathematical language and symbols;realize that representing, discussing, reading, writing, and listening to mathematics are a vital part of learning and using mathematics.

Mathematics can be thought of as a language that must be meaningful if students are to communicate mathematically and apply mathematics productively. Communication plays an important role in helping children construct links between their informal, intuitive notions and the abstract language and symbolism of mathematics; it also plays a key role in helping children make important connections among physical, pictorial, graphic, symbolic, verbal, and mental representations of mathematical ideas. When children see that one representation, such as an equation, can describe many situations, they begin to understand the power of mathematics; when they realize that some ways of representing a problem are more helpful than others, they begin to understand the flexibility and usefulness of mathematics.

NCTM Math Standards K-4

STANDARD 3: MATHEMATICS AS REASONING |
Examples of Reasoning |

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In grades K--4, the study of mathematics should emphasize reasoning so that students can--

draw logical conclusions about mathematics;use models, known facts, properties, and relationships to explain their thinking;justify their answers and solution processes;use patterns and relationships to analyze mathematical situations;believe that mathematics makes sense.

A major goal of mathematics instruction is to help children develop the belief that they have the power to do mathematics and that they have control over their own success or failure. This autonomy develops as children gain confidence in their ability to reason and justify their thinking. It grows as children learn that mathematics is not simply memorizing rules and procedures but that mathematics makes sense, is logical, and is enjoyable. A classroom that values reasoning also values communicating and problem solving, all of which are components of the broad goals of the entire elementary school curriculum.

NCTM Math Standards K-4

STANDARD 4: MATHEMATICAL CONNECTIONS |
Examples of Math Connections |

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In grades K--4, the study of mathematics should include opportunities to make connections so that students can--

link conceptual and procedural knowledge;relate various representations of concepts or procedures to one another;recognize relationships among different topics in mathematics;use mathematics in other curriculum areas;use mathematics in their daily lives.

This standard's purpose is to help children see how mathematical ideas are related. The mathematics curriculum is generally viewed as consisting of several discrete strands. As a result, computation, geometry, measurement, and problem solving tend to be taught in isolation. It is important that children connect ideas both among and within areas of mathematics. Without such connections, children must learn and remember too many isolated concepts and skills rather than recognizing general principles relevant to several areas. When mathematical ideas are also connected to everyday experiences, both in and out of school, children become aware of the usefulness of mathematics.

NCTM Math Standards K-4

STANDARD 5: ESTIMATION |
Examples of Estimation |

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In grades K-4, the curriculum should include estimation so students can--

explore estimation strategies;recognize when an estimate is appropriate;determine the reasonableness of results;apply estimation in working with quantities, measurement, computation, and problem solving.

Estimation presents students with another dimension of mathematics; terms such as *about,
near, closer to, between*, and *a little less than* illustrate that mathematics
involves more than exactness. Estimation interacts with number sense and spatial sense to
help children develop insights into concepts and procedures, flexibility in working with
numbers and measurements, and an awareness of reasonable results. Estimation skills and
understanding enhance the abilities of children to deal with everyday quantitative
situations.

NCTM Math Standards K-4

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STANDARD 6: NUMBER SENSE AND NUMERATION |
Examples of Number Sense, and Numeration |

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In grades K-4, the mathematics curriculum should include whole number concepts and skills so that students can--

construct number meanings through real-world experiences and the use of physical materials;understand our numeration system by relating counting, grouping, and place-value concepts;develop number sense;interpret the multiple uses of numbers encountered in the real world.

Children must understand numbers if they are to make sense of the ways numbers are used in their everyday world. They need to use numbers to quantify, to identify location, to identify a specific object in a collection, to name, and to measure. Furthermore, an understanding of place value is crucial for later work with number and computation.

NCTM Math Standards K-4

STANDARD7: CONCEPTS OF WHOLE NUMBER OPERATIONS |
Examples of Whole Number Operations |

In grades K-4, the mathematics curriculum should include concepts of addition, subtraction, multiplication, and division of whole numbers so that students can--

develop meaning for the operations by modeling and discussing a rich variety of problem situations;relate the mathematical language and symbolism of operations to problem situations and informal language;recognize that a wide variety of problem structures can be represented by a single operation;develop operation sense.

Understanding the fundamental operations of addition, subtraction, multiplication, and
division is central to knowing mathematics. One essential component of what it means to
understand an operation is recognizing conditions in real-world situations that indicate
that the operation would be useful in those situations. Other components include building
an awareness of models and the properties of an operation, seeing relationships among
operations, and acquiring insight into the effects of an operation on a pair of numbers.
These four components are aspects of *operation sense*. Children with good operation
sense are able to apply operations meaningfully and with flexibility. Operation sense
interacts with number sense and enables students to make thoughtful decisions about the
reasonableness of results. Furthermore, operation sense provides a framework for the
conceptual development of mental and written computational procedures.

NCTM Math Standards K-4

STANDARD 8: WHOLE NUMBER COMPUTATION |
Examples of Whole Number Computation |

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In grades K-4, the mathematics curriculum should develop whole number computation so that students can--

model, explain, and develop reasonable proficiency with basic facts and algorithms;use a variety of mental computation and estimation techniques;use calculators in appropriate computational situations;select and use computation techniques appropriate to specific problems and determine whether the results are reasonable.

The purpose of computation is to solve problems. Thus, although computation is important in mathematics and in daily life, our technological age requires us to rethink how computation is done today. Almost all complex computation today is done by calculators and computers. In many daily situations, answers are computed mentally or estimates are sufficient, and paper-and-pencil algorithms are useful when the computation is reasonably straightforward. This standard addresses the importance of teaching children a variety of ways to compute, as well as the usefulness of calculators in solving problems containing large numbers or requiring complex computations. Related to this goal is the necessity of having reasonable expectations for proficiency with paper-and-pencil computation. Clearly, paper-and-pencil computation cannot continue to dominate the curriculum or there will be insufficient time for children to learn other, more important mathematics they need now and in the future.

NCTM Math Standards K-4

STANDARD 9: GEOMETRY AND SPATIAL SENSE |
Examples of Geometry and Spatial Sense |

In grades K-4, the mathematics curriculum should include two- and three-dimensional geometry so that students can--

describe, model, draw, and classify shapes;investigate and predict the results of combining, subdividing, and changing shapes;develop spatial sense;relate geometric ideas to number and measurement ideas;recognize and appreciate geometry in their world.

Geometry is an important component of the K-4 mathematics curriculum because geometric knowledge, relationships, and insights are useful in everyday situations and are connected to other mathematical topics and school subjects. Geometry helps us represent and describe in an orderly manner the world in which we live. Children are naturally interested in geometry and find it intriguing and motivating; their spatial capabilities frequently exceed their numerical skills, and tapping these strengths can foster an interest in mathematics and improve number understandings and skills.

NCTM Math Standards K-4
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STANDARD 10: MEASUREMENT |
Examples of Measurement |

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In grades K-4, the mathematics curriculum should include measurement so that students can-

understand the attributes of length, capacity, weight, mass, area, volume, time, temperature, and angle;develop the process of measuring and concepts related to units of measurement;make and use estimates of measurement;make and use measurements in problem and everyday situations.

Measurement is of central importance to the curriculum because of its power to help children see that mathematics is useful in everyday life and to help them develop many mathematical concepts and skills. Measuring is a natural context in which to introduce the need for learning about fractions and decimals, and it encourages children to be actively involved in solving and discussing problems.

NCTM Math Standards K-4

STANDARD 11: STATISTICS
AND PROBABILITY |
Examples of Statistics and Probability |

In grades K-4, the mathematics curriculum should include experiences with data analysis and probability so that students can--

collect, organize, and describe data;construct, read, and interpret displays of data;formulate and solve problems that involve collecting and analyzing data;explore concepts of chance.

Collecting, organizing, describing, displaying, and interpreting data, as well as making decisions and predictions on the basis of that information, are skills that are increasingly important in a society based on technology and communication. These processes are particularly appropriate for young children because they can be used to solve problems that often are inherently interesting, represent significant, applications of mathematics to practical questions, and offer rich opportunities for mathematical inquiry. The study of statistics and probability highlights the importance of questioning, conjecturing, and searching for relationships when formulating and solving real-world problems.

NCTM Math Standards K-4

STANDARD 12: FRACTIONS AND DECIMALS |
Examples of Fractions and Decimals |

In grades K-4, the mathematics curriculum should include fractions and decimals so that students can--

develop concepts of fractions, mixed numbers, and decimals;develop number sense for fractions and decimals;use models to relate fractions to decimals and to find equivalent fractions;use models to explore operations on fractions and decimals;apply fractions and decimals to problem situations.

Fractions and decimals represent a significant extension of children's knowledge about numbers. When children possess a sound understanding of fraction and decimal concepts, they can use this knowledge to describe real-world phenomena and apply it to problems involving measurement, probability, and statistics. An understanding of fractions and decimals broadens students' awareness of the usefulness and power of numbers and extends their knowledge of the number system. It is critical in grades K-4 to develop concepts and relationships that will serve as a foundation for more advanced concepts and skills.

NCTM Math Standards K-4

STANDARD 13: PATTERNS AND RELATIONSHIPS |
Examples of Patterns |

In grades K-4, the mathematics curriculum should include the study of patterns and relationships so that student can--

recognize, describe, extend, and create a wide variety of patterns;represent and describe mathematical relationships;explore the use of variables and open sentences to express relationships.

Patterns are everywhere. Children who are encouraged to look for patterns and to express them mathematically begin to understand how mathematics applies to the world in which they live. Identifying and working with a wide variety of patterns help children to develop the ability to classify and organize information. Relating patterns in numbers, geometry, and measurement helps them understand connections among mathematical topics. Such connections foster the kind of mathematical thinking that serves as a foundation for the more abstract ideas studied in later grades.

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**Updated: December 20, 2004**