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Excerpts from the National Council of Teachers of Mathematics Standards

The Standards of the National Council of Teachers of Mathematics emphasizes that patterns are of recurring importance to mathematics. The following is excerpted from the NCTM's Curriculum and Evaluation Standards for School Mathematics. Reston, VA: NCTM.

Grades K-4: Standard 13:
Patterns and Relations

p. 60 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 helps 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.

From the earliest grades, the curriculum should give students opportunities to focus on regularities in events, shapes, designs, and sets of numbers. Children should begin to see that regularity is the essence of mathematics. The idea of a functional relationship can be intuitively developed through observations of regularity and work with generalizable patterns.

Physical materials and pictorial displays should be used to help children recognize and create patterns and relationships. Observing varied representations of the same pattern helps children identify its properties. The use of letters and other symbols in generalizing descriptions of these properties prepares children to use variables in the future. This experience builds readiness for a generalized view of mathematics and a later study of algebra....

p. 61 Pattern recognition involves many concepts, such as color and shape identification, direction, orientation, size, and number relationships. Children should use all these properties in identifying, extending, and creating patterns. Identifying the "cores" of patterns helps children become aware of the structures. For example, in some patterns the core repeats, whereas in others, the core grows.

Grades 5-8: Standard B:
Patterns and Functions

p. 98 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....Exploring patterns helps students develop mathematical power and instills in them an appreciation for the beauty of mathematics.

The study of patterns in grades 5-8 builds on students' experiences in K-4 but shifts emphasis to an exploration of functions. However, work with patterns continues to be informal and relatively unburdened by symbolism. Students have opportunities to generalize and describe patterns and functions in many ways and to explore the relationships among them....

During the middle years, the study of patterns and functions should focus on the analysis, representation, and generalization of functional relationships. These topics should first be explored as informal investigations.

Students should be encouraged to observe and describe all sorts of patterns in the world around them: plowed fields, haystacks, architecture, paintings, leaves on trees, spirals on pineapples, and so on....

p. 99 Looking for patterns in simple situations can lead to a method of counting generalizable in other situations....

Excerpts from the National Standards for Arts Educators

Consortium of National Arts Education Associations (1994). National Standards for Arts Education: What Every Young American Should Know and Be Able to Do in the Arts. Reston, VA: Music Educators National Conference.


p. 98 The Benefits of Arts Education. Arts education benefits the student because it cultivates the whole child, gradually building many kinds of literacy while developing intuition, reasoning, imagination, and dexterity into unique forms of expression and communication. This process requires not merely an active mind but a trained one. An education in the arts benefits society because students of the arts gain powerful tools for understanding human experiences both past and present. They learn to respect the often very different ways others have of thinking, working, and expressing themselves. They learn to make decisions in situations where there are no standard answers. By studying the arts, students stimulate their natural creativity and learn to develop it to meet the needs of a complex and competitive society. And, as study and competence in the arts reinforce one [another], the joy of learning becomes real, tangible, and powerful.

The Arts and Other Core Subjects. The Standards address competence in the arts disciplines first of all. But that competence provides a firm foundation for connecting arts-related concepts and facts across the art forms and from them to the sciences and humanities. For example, the intellectual methods of the arts are precisely those used to transform scientific disciplines and discoveries into everyday technology.


p. 98 Children in grades K-4 love to move and learn through engagement of the whole self. They need to become literate in the language of dance in order to use this natural facility as a means of communication and self-expression, and as a way of responding to the expression of others....Students learn basic movement and choreographic skills in musical/rhythmic contexts.... The skills and knowledge acquired allow them to begin working independently and with a partner in creating and performing dances.

Experiences in perceiving and responding to dance expand students' vocabularies, enhance their listening and viewing skills, and enable them to think critically.... They investigate questions such as "What is it? How does it work? Why is it important?" ....Students learn to compare works in terms of the elements of space, time, and force/ energy and to experience the similarities and differences between dance and other disciplines.

Through dance education, students can also come to an understanding of their own culture and to begin to respect dance as part of the heritage of many cultures....

p. 39 Students in grades 5-8 develop a sense of themselves in relation to others and in relation to the world. As a result, they are ready to respond more thoughtfully to dance, to perceive details of style and choreographic structure, and to reflect upon what is communicated.

p. 55 High school students need to continue to dance and create dances in order to develop more highly their ability to communicate in a way that is different from the written or spoken word, or even from other visual or auditory symbol systems. They also need to respect their bodies and to understand that dance is the product of intentional and intelligent physical actions....Because dance involves abstract images, students can develop higher order thinking skills through perceiving, analyzing, and making discriminating judgments about dance.


p. 42 The period represented by grades 5-8 is especially critical in students' musical development.... Composing and improvising provide students with unique insight into the form and structure of music and at the same time help them to develop their creativity. Broad experience with a variety of music is necessary if students are to make informed musical judgments. Similarly, this breadth of background enables them to begin to understand the connections and relationships between music and other disciplines.

p. 59 Through singing, playing instruments, and composing, students can express themselves creatively, while a knowledge of notation and performance traditions enables them to learn new music independently throughout their lives. Skills in analysis, evaluation, and synthesis are important because they enable students to recognize and pursue excellence in their musical experiences and to understand and enrich their environment.

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