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.
K-4: Standard 13:
Patterns and Relations
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.
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
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....
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.
5-8: Standard B:
Patterns and Functions
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.
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
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
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....
99 Looking for patterns in simple situations can lead to a method
of counting generalizable in other situations....
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
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,
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
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.
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.
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....
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.
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.
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.
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