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NCTM Math Standards 5-8
STANDARD 1: MATHEMATICS AS PROBLEM SOLVING | Examples of Problem Solving |
In grades 5-8, the study of mathematics should emphasize problem solving so that students can--
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.
NCTM Math Standards 5-8
STANDARD 2: MATHEMATICS AS COMMUNICATION | Examples of Communication |
In grades 5-8, the study of mathematics should include opportunities to communicate so that students can--
- model situations using oral, written, concrete, pictorial, graphical, and algebraic methods;
- reflect on and clarify their own thinking about mathematical ideas and situations;
- develop common understandings of mathematical ideas, including the role of definitions;
- use the skills of reading, listening, and viewing to interpret and evaluate mathematical ideas;
- discuss mathematical ideas and make conjectures and convincing arguments;
- appreciate the value of mathematical notation and its role in the development of mathematical ideas.
The use of mathematics in other disciplines has increased dramatically, largely because of its power to represent and communicate ideas concisely. Society's increasing use of technology requires that students learn both to communicate with computers and to make use of their own individual power as a medium of communication. The ability to read, write, listen, think creatively, and communicate about problems will develop and deepen students' understanding of mathematics.
NCTM Math Standards 5-8
STANDARD 3: MATHEMATICS AS REASONING | Examples of Reasoning |
In grades 5-8, reasoning shall permeate the mathematics curriculum so that students can--
- recognize and apply deductive and inductive reasoning;
- understand and apply reasoning processes, with special attention to spatial reasoning and reasoning with proportions and graphs;
- make and evaluate mathematical conjectures and arguments;
- validate their own thinking;
- appreciate the pervasive use and power of reasoning as a part of mathematics.
Reasoning is fundamental to the knowing and doing of mathematics. Although most disciplines have standards of evaluation by which new theories or discoveries are judged, nowhere are these standards as explicit and well formulated as they are in mathematics. Conjecturing and demonstrating the logical validity of conjectures are the essence of the creative act of doing mathematics. To give more students access to mathematics as a powerful way of making sense of the world, it is essential that an emphasis on reasoning pervade all mathematical activity. Students need a great deal of time and many experiences to develop their ability to construct valid arguments in problem settings and evaluate the arguments of others.
NCTM Math Standards 5-8
STANDARD 4: MATHEMATICAL CONNECTIONS | Examples of Connections |
In grades 5-8, the mathematics curriculum should include the investigation of mathematical connections so that students can--
- see mathematics as an integrated whole;
- explore problems and describe results using graphical, numerical, physical, algebraic, and verbal mathematical models or representations;
- use a mathematical idea to further their understanding of other mathematical ideas;
- apply mathematical thinking and modeling to solve problems that arise in other disciplines, such as art, music, psychology, science, and business;
- value the role of mathematics in our culture and society.
For many students, mathematics in the middle grades has far too often simply repeated or extended much of the computational work covered in the earlier grades. The intent of this standard is to help students broaden their perspective, to view mathematics as an integrated whole rather than as an isolated set of topics, and to acknowledge its relevance and usefulness both in and out of school. Mathematics instruction at the 5-8 level should prepare students for expanded and deeper study in high school through exploration of the interconnections among mathematical ideas.
NCTM Math Standards 5-8
STANDARD 5: NUMBER AND NUMBER RELATIONSHIPS | Examples of Number Relations |
In grades 5-8, the mathematics curriculum should include the continued development of number and number relationships so that students can--
- understand, represent, and use numbers in a variety of equivalent forms (integer, fraction, decimal, percent, exponential, and scientific notation) in real-world and mathematical problem situations;
- develop number sense for whole numbers, fractions, decimals, integers, and rational numbers;
- understand and apply ratios, proportions, and percents in a wide variety of situations;
- investigate relationships among fractions, decimals, and percents;
- represent numerical relationships in one- and two-dimensional graphs.
The use of concise symbols and language to represent numbers is a significant historical and practical development. In the middle school years, students come to recognize that numbers have multiple representations, so the development of concepts for fractions, ratios, decimals, and percents and the idea of multiple representations of these numbers need special attention and emphasis. The ability to generate, read, use, and appreciate multiple representations of the same quantity is a critical step in learning to understand and do mathematics.
NCTM Math Standards 5-8
STANDARD 6: NUMBER SYSTEMS AND NUMBER THEORY | Examples of Number Sense, Concepts, and Operations |
In grades 5-8, the mathematics curriculum should include the study of number systems and number theory so that students can--
- understand and appreciate the need for numbers beyond the whole numbers;
- develop and use order relations for whole numbers, fractions, decimals, integers, and rational numbers;
- extend their understanding of whole number operations to fractions, decimals, integers, and rational numbers;
- understand how the basic arithmetic operations are related to one another;
- develop and apply number theory concepts (e.g., primes, factors, and multiples) in real-world and mathematical problem situations.
The central theme of this standard is the underlying structure of mathematics, which bonds its many individual facets into a useful, interesting, and logical whole. Instruction in grades 5-8 typically devotes a great deal of time to helping students master a myriad of details but pays scant attention to how these individual facets fit together. It is the intent of this standard that students should come to understand and appreciate mathematics as a coherent body of knowledge rather than a vast, perhaps bewildering, collection of isolated facts and rules. Understanding this structure promotes students' efficiency in investigating the arithmetic of fractions, decimals, integers, and rationals through the unity of common ideas. It also offers insights into how the whole number system is extended to the rational number system and beyond. It improves problem-solving capability by providing a better perspective of arithmetic operations.
NCTM Math Standards 5-8
STANDARD 7: COMPUTATION AND ESTIMATION | Examples of Estimation |
In grades 5-8, the mathematics curriculum should develop the concepts underlying computation and estimation in various contexts so that students can--
- compute with whole numbers, fractions, decimals, integers, and rational numbers;
- develop, analyze, and explain procedures for computation and techniques for estimation;
- develop, analyze, and explain methods for solving proportions;
- select and use an appropriate method for computing from among mental arithmetic, paper-and-pencil, calculator, and computer methods;
- use computation, estimation, and proportions to solve problems;
- use estimation to check the reasonableness of results.
Although computation is vital in this information age, technology has drastically changed the methods by which we compute. Whereas inexpensive calculators execute routine computations accurately and quickly and computers execute more complex computations with ease, many current mathematics programs focus on traditional paper-and-pencil algorithms. This standard prepares students to select and use appropriate mental, paper-and-pencil, calculator, and computer methods.
NCTM Math Standards 5-8
STANDARD 8: PATTERNS AND FUNCTIONS | Examples of Patterns and Functions |
In grades 5-8, the mathematics curriculum should include explorations of patterns and functions so that students can--
- describe, extend, analyze, and create a wide variety of patterns;
- describe and represent relationships with tables, graphs, and rules;
- analyze functional relationships to explain how a change in one quantity results in a change in another;
- use patterns and functions to represent and solve problems.
One of the central themes of mathematics is the study of patterns and functions. This study requires students to recognize, describe, and generalize patterns and build mathematical models to predict the behavior of real-world phenomena that exhibit the observed pattern. The widespread occurrence of regular and chaotic pattern behavior makes the study of patterns and functions important. Exploring patterns helps students develop mathematical power and instills in them an appreciation for the beauty of mathematics.
NCTM Math Standards 5-8
STANDARD 9: ALGEBRA |
In grades 5-8, the mathematics curriculum should include explorations of algebraic concepts and processes so that students can-
- understand the concepts of variable, expression, and equation;
- represent situations and number patterns with tables, graphs, verbal rules, and equations and explore the interrelationships of these representations;
- analyze tables and graphs to identify properties and relationships;
- develop confidence in solving linear equations using concrete, informal, and formal methods;
- investigate inequalities and nonlinear equations informally;
- apply algebraic methods to solve a variety of real-world and mathematical problems.
The middle school mathematics curriculum is, in many ways, a bridge between the concrete elementary school curriculum and the more formal mathematics curriculum of the high school. One critical transition is that between arithmetic and algebra. It is thus essential that in grades 5-8, students explore algebraic concepts in an informal way to build a foundation for the subsequent formal study of algebra. Such informal explorations should emphasize physical models, data, graphs, and other mathematical representations rather than facility with formal algebraic manipulation. Students should be taught to generalize number patterns to model, represent, or describe observed physical patterns, regularities, and problems. These informal explorations of algebraic concepts should help students to gain confidence in their ability to abstract relationships from contextual information and use a variety of representations to describe those relationships.
NCTM Math Standards 5-8
STANDARD 10: STATISTICS | Examples of Statistics |
In grades 5-8, the mathematics curriculum should include exploration of statistics in real-world situations so that students can--
- systematically collect, organize, and describe data;
- construct, read, and interpret tables, charts, and graphs;
- make inferences and convincing arguments that are based on data analysis;
- evaluate arguments that are based on data analysis;
- develop an appreciation for statistical methods as powerful means for decision making.
In this age of information and technology, an ever-increasing need exists to understand how information is processed and translated into usable knowledge. Because of society's expanding use of data for prediction and decision making, it is important that students develop an understanding of the concepts and processes used in analyzing data. A knowledge of statistics is necessary if students are to become intelligent consumers who can make critical and informed decisions.
NCTM Math Standards 5-8
STANDARD 11: PROBABILITY | Examples of Probability |
In grades 5-8, the mathematics curriculum should include explorations of probability in real-world situations so that students can--
- model situations by devising and carrying out experiments or simulations to determine probabilities;
- model situations by constructing a sample space to determine probabilities;
- appreciate the power of using a probability model by comparing experimental results with mathematical expectations;
- make predictions that are based on experimental or theoretical probabilities;
- develop an appreciation for the pervasive use of probability in the real world.
Probability theory is the underpinning of the modern world. Current research in both the physical and social sciences cannot be understood without it. Today's politics, tomorrow's weather report and next week's satellites all depend on it. (Huff and Geise 1959)
An understanding of probability and the related area of statistics is essential to being an informed citizen. Often we read statements such as, "There is a 20 percent chance of rain or snow today." "The odds are three to two that the Cats will win the championship." "The probability of winning the grand prize in the state lottery is 1 in 7 240 000." Students in the middle grades have an intense interest in the notions of fairness and the chances of winning games. The study of probability develops concepts and methods for investigating such situations. These methods allow students to make predictions when uncertainty exists and to make sense of claims that they see and hear.
NCTM Math Standards 5-8
STANDARD 12: GEOMETRY | Examples of Geometry and Spatial Sense |
In grades 5-8, the mathematics curriculum should include the study of the geometry of one, two, and three dimensions in a variety of situations so that students can--
- identify, describe, compare, and classify geometric figures;
- visualize and represent geometric figures with special attention to developing spatial sense;
- explore transformations of geometric figures;
- represent and solve problems using geometric models;
- understand and apply geometric properties and relationships;
- develop an appreciation of geometry as a means of describing the physical world.
Geometry is grasping space . . . that space in which the child lives, breathes and moves. The space that the child must learn to know, explore, conquer, in order to live, breathe and move better in it. (Freudenthal 1973, p. 403).
The study of geometry helps students represent and make sense of the world. Geometric models provide a perspective from which students can analyze and solve problems, and geometric interpretations can help make an abstract (symbolic) representation more easily understood. Many ideas about number and measurement arise from attempts to quantify real-world objects that can be viewed geometrically. For example, the use of area models provides an interpretation for much of the arithmetic of decimals, fractions, ratios, proportions, and percents.
NCTM Math Standards 5-8
STANDARD 13: MEASUREMENT | Examples of Measurement |
In grades 5-8, the mathematics curriculum should include extensive concrete experiences using measurement so that students can--
- extend their understanding of the process of measurement;
- estimate, make, and use measurements to describe and compare phenomena;
- select appropriate units and tools to measure to the degree of accuracy required in a particular situation;
- understand the structure and use of systems of measurement;
- extend their understanding of the concepts of perimeter, area, volume, angle measure, capacity, and weight and mass;
- develop the concepts of rates and other derived and indirect measurements;
- develop formulas and procedures for determining measures to solve problems.
Measurement activities can and should require a dynamic interaction between students and their environment. Students encounter measurement ideas both in and out of school, in such areas as architecture, art, science, commercial design, sports, cooking, shopping, and map reading. The study of measurement shows the usefulness and practical applications of mathematics, and students' need to communicate about various measurements highlights the importance of standard units and common measurement systems.
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Updated: March 12, 2004