Standard 5: Number and Number Relations
from the NCTM Standards
In grades 5-8 the mathematics curriculum should include the continued
development of number and number relationships so that students
can understand and apply ratios, proportions, and percents in
a wide variety of situations.
In the middle school years, students come to recognize that numbers
have multiple representations, so the development of concepts for
fractions, ratios, decimals, and percentages and the idea of multiple
representation of these numbers needs 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. To provide students with a lasting sense of
number and number relationships, learning should be grounded in
experience related to aspects of everyday life or to the use of
concrete materials designed to reflect underlying mathematical ideas.
Students should encounter number lines, area models, and graphs
as well as representations of number that appear on calculators
and computers (e.g., forms of scientific notation). Students should
learn to identify equivalent forms of a number and understand why
a particular representation is useful in a given setting (page 87).
Standard 13: Measurement in the Middle School
from the NCTM Standards
In grades 5-8 the mathematics curriculum should include extensive
concrete experiences using measurements so that students can
- extend their understanding of the concept of perimeter, area,
volume, angle measure, capacity, and weight and mass
- develop formulas and procedures for determining measures to
The development of the concepts of perimeter, area, volume, angle
measure, capacity, and weight is initiated in grades K-4 and extended
and applied in grades 5-8. From their exploration, students should
develop multiplicative procedures and formulas for determining measures.
The curriculum should focus on the development of understanding,
not on the rote memorization of formulas (page 116).
As students progress through grades 5-8, they should develop more
efficient procedures and, ultimately, formulas for finding measures.
Length, area, and volume of one-, two- and three-dimensional figures
are especially important over these grade levels (page 118).
Reprinted with permission from Curriculum and Evaluation Standards
for School Mathematics. Order from NCTM, 1900 Association Drive,
Reston, VA 22091. Telephone: 1-800-235-7566.
Science Education Program Standards
Excerpts from the National Science Education Standards
The program standards are criteria for the quality and conditions
for school science programs. They focus on issues at the school
and district levels that relate to opportunities for students to
learn and opportunities for teachers to teach science.
In an effective science program, a set of clear goals and expectations
for students must be used to guide the design, implementation, and
assessment of all elements of the science program. A science program
begins with the goals and expectations for student achievement;
it also includes the selection and organization of science content
into curriculum frameworks, ways of teaching, and assessment strategies.
The goals for a science program provide the statements of philosophy
and the vision that drives the program and the statements of purpose
that the program is designed to achieve.
Curriculum frameworks should be used to guide the selection and
development of units and courses of study. The curriculum framework
provides a guide for moving the vision presented in the goals closer
to reality. Teachers use the guide as they select and design specific
school and classroom work. By specifying the sequence of topics
in the curriculum, the guide ensures articulation and coherence
across the curriculum. Using the framework, teachers design instruction
that is based on the prior experiences of students but avoids unnecessary
repetition. The framework guides the students as they move through
Teaching practices need to be consistent with the goals and curriculum
frameworks. The program standards do not prescribe specific teaching
behaviors, nor should district or school policies. There are many
ways to teach science effectively while adhering to the basic tenets
of the National Science Education Standards, but they must be consistent
with the goals and framework of the district.
Support systems and formal and informal expectations of teachers
must be aligned with the goals, student expectations, and curriculum
frameworks. An effective science program requires an adequate support
system, including resources of people, time, materials and finance,
opportunities for staff development, and leadership that works toward
the goals of the program. It is encoded formally in policy documents
such as a teacher's handbook and informally in the unwritten norms
that determine routines. The support system must support classroom
teachers in teaching science as described in the Standards (page
The above excerpts are reprinted, by permission of the National
Academy Press, from the National Science Education Standards.
Copyright 1996 by the National Academy of Sciences, courtesy of
the National Academy Press, Washington, D.C.