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This section presents
thirteen curriculum standards for grades 5-8:
- Mathematics
as Problem Solving
- Mathematics
as Communication
- Mathematics
as Reasoning
- Mathematical
Connections
- Number
and Number Relationships
- Number
Systems and Number Theory
- Computation
and Estimation
- Patterns
and Functions
- Algebra
- Statistics
- Probability
- Geometry
- Measurement
Mathematics is a useful,
exciting, and creative area of study that can be appreciated and
enjoyed by all students in grades 5-8. It helps them develop their
ability to solve problems and reason logically. It offers to these
curious, energetic students a way to explore and make sense of their
world. However, many students view the current mathematics curriculum
in grades 5-8 as irrelevant, dull, and routine. Instruction has
emphasized computational facility at the expense of a broad, integrated
view of mathematics and has reflected neither the vitality of the
subject nor the characteristics of the students.
An ideal
5-8 mathematics curriculum would expand students' knowledge of numbers,
computation, estimation, measurement, geometry, statistics, probability,
patterns and functions, and the fundamental concepts of algebra.
The need for this kind of broadened curriculum is acute. An examination
of textbook series shows the repetition of topics, approach, and
level of presentation in grade after grade. A comparison of the
tables of contents shows little change over grades 5-8. It is even
more disconcerting to realize that the very chapters that contain
the most new material, such as probability, statistics, geometry,
and prealgebra, are covered in the last half of the books--the sections
most often skipped by teachers for lack of time. The result is an
ineffective curriculum that rehashes material students already have
seen. Such a curriculum promotes a negative image of mathematics
and fails to give students an adequate background for secondary
school mathematics.
These thirteen standards
promote a broad curriculum for students in grades 5-8. Developing
certain computational skills is important but constitutes only a
part of this curriculum. Nevertheless, the existing curriculum in
some schools prohibits many students from studying a broader curriculum
until they have "mastered" basic computational skills.
Shifting the focus to a broader curriculum is important for the
following reasons:
- Basic skills today and
in the future mean far more than computational proficiency. Moreover,
the calculator renders obsolete much of the complex paper-and-pencil
proficiency traditionally emphasized in mathematics courses. Topics
such as geometry, probability, statistics, and algebra have become
increasingly more important and accessible to students through
technology.
- If students have not
been successful in "mastering" basic computational skills
in previous years, why should they be successful now, especially
if the same methods that failed in the past are merely repeated?
In fact, considering the effect of failure on students' attitudes,
we might argue that further efforts toward mastering computational
skills are counterproductive.
- Many of the mathematics
topics that are omitted actually can help students recognize the
need for arithmetic concepts and skills and provide fresh settings
for their use. For example, in probability, students have many
opportunities to add and multiply fractions.
The vision articulated
in the 5-8 standards is of a broad, concept-driven curriculum, one
that reflects the full breadth of relevant mathematics and its interrelationships
with technology. This vision is built on five overall curricular
goals for students: learning to value mathematics, becoming confident
in their ability, becoming a mathematical problem solver, learning
to communicate mathematically, and learning to reason mathematically.
The teaching of this curriculum should be related to the characteristics
of middle school students and their current and future needs.
The 5-8 curriculum should
include the following features:
- Problem situations that
establish the need for new ideas and motivate students should
serve as the context for mathematics in grades 5-8. Although a
specific idea might be forgotten, the context in which it is learned
can be remembered and the idea re-created. In developing the problem
situations, teachers should emphasize the application of mathematics
to real-world problems as well as to other settings relevant to
middle school students.
- Communication with and
about mathematics and mathematical reasoning should permeate the
5-8 curriculum.
- A broad range of topics
should be taught, including number concepts, computation, estimation,
functions, algebra, statistics, probability, geometry, and measurement.
Although each of these areas is valid mathematics in its own right,
they should be taught as an integrated whole, not as isolated
topics; the connections among them should be a prominent feature
of the curriculum.
- Technology, including
calculators, computers, and videos, should be used when appropriate.
These devices and formats free students from tedious computations
and allow them to concentrate on problem solving and other important
content. They also give them new means to explore content. As
paper-and-pencil computation becomes less important, the skills
and understanding required to make proficient use of calculators
and computers become more important.
The standards are not intended
to each constitute a chapter in a text or a particular unit of instruction;
rather, learning activities should incorporate topics and ideas
across standards. For example, an instructional activity might involve
problem solving and use geometry, measurement, and computation.
All mathematics should be studied in contexts that give the ideas
and concepts meaning. Problems should arise from situations that
are not always well formed. Students should have opportunities to
formulate problems and questions that stem from their own interests.
Learning should engage
students both intellectually and physically. They must become active
learners, challenged to apply their prior knowledge and experience
in new and increasingly more difficult situations. Instructional
approaches should engage students in the process of learning rather
than transmit information for them to receive. Middle grade students
are especially responsive to hands-on activities in tactile, auditory,
and visual instructional modes.
Classroom activities should
provide students the opportunity to work both individually and in
small- and large-group arrangements. The arrangement should be determined
by the instructional goals as well as the nature of the activity.
Individual work can help students develop confidence in their own
ability to solve problems but should constitute only a portion of
the middle school experience. Working in small groups provides students
with opportunities to talk about ideas and listen to their peers,
enables teachers to interact more closely with students, takes positive
advantage of the social characteristics of the middle school student,
and provides opportunities for students to exchange ideas and hence
develops their ability to communicate and reason. Small-group work
can involve collaborative or cooperative as well as independent
work. Projects and small-group work can empower students to become
more independent in their own learning. Whole-class discussions
require students to synthesize, critique, and summarize strategies,
ideas, or conjectures that are the products of individual and group
work. These mathematical ideas can be expanded to, and integrated
with, other subjects.
The 5-8 standards make
the following assumptions about classroom materials:
- Every classroom will
be equipped with ample sets of manipulative materials and supplies
(e.g., spinners, cubes, tiles, geoboards, pattern blocks, scales,
compasses, scissors, rulers, protractors, graph paper, grid-and-dot
paper).
- Teachers and students
will have access to appropriate resource materials from which
to develop problems and ideas for explorations.
- All students will have
a calculator with functions consistent with the tasks envisioned
in this curriculum. Calculators should include the following features:
algebraic logic including order of operations; computation in
decimal and common fraction form; constant function for addition,
subtraction, multiplication, and division; and memory, percent,
square root, exponent, reciprocal, and +/- keys.
- Every classroom will
have at least one computer available at all times for demonstrations
and student use. Additional computers should be available for
individual, small-group, and whole-class use.
Implementation of the 5-8
standards should consider the unique characteristics of middle school
students. As vast changes occur in their intellectual, psychological,
social, and physical development, students in grades 5-8 begin to
develop their abilities to think and reason more abstractly. Throughout
this period, however, concrete experiences should continue to provide
the means by which they construct knowledge. From these experiences
they abstract more complex meanings and ideas. The use of language,
both written and oral, helps students clarify their thinking and
report their observations as they form and verify their mathematical
ideas.
Students at this level
can aptly be called "children in transition": they are
restless, energetic, responsive to peer influence, and unsure about
themselves. Self-consciousness is their hallmark, and curiosity
about such questions as Who am I? How do I fit in? What do I enjoy
doing? What do I want to be? is both their motivation and their
nemesis. From this turmoil emerges an individual, with attitudes
and patterns of thought taking shape.
In the transition to adulthood,
middle school students are forming lifelong values and skills. The
decisions students make about what they will study and how they
will learn can dramatically affect their future. Failure to study
mathematics can close the doors to vocational-technical schools,
college majors, and careers--a loss of opportunity that happens
most often to young women and minority students. Because many of
the attitudes that affect these decisions are developed during the
middle grades, it is crucial that conscious efforts be made to encourage
all students, especially young women and minorities, to pursue mathematics.
To this end, the curriculum must be interesting and relevant, must
emphasize the usefulness of mathematics, and must foster a positive
disposition toward mathematics.
Whenever possible, students'
cultural backgrounds should be integrated into the learning experience.
Black or Hispanic students, for example, may find the development
of mathematical ideas in their cultures of great interest. Teachers
must also be sensitive to the fact that students bring very different
everyday experiences to the mathematics classroom. The way in which
a student from an urban environment and a student from a suburban
or rural environment interpret a problem situation can be very different.
This is an important reason why communication is one of the overarching
goals of these standards.
Students will perform better
and learn more in a caring environment in which they feel free to
explore mathematical ideas, ask questions, discuss their ideas,
and make mistakes. By listening to students' ideas and encouraging
them to listen to one another, one can establish an atmosphere of
mutual respect. Teachers can foster this willingness to share by
helping students explore a variety of ideas in reaching solutions
and verifying their own thinking. This approach instills in students
an understanding of the value of independent learning and judgment
and discourages them from relying on an outside authority to tell
them whether they are right or wrong.
Because the curriculum,
activities, and mathematical knowledge envisioned in these standards
are conceptually based, evaluation is not a simple or narrow task.
The development of conceptual understanding is a long-term process;
understanding is developed, elaborated, deepened, and made more
nearly complete over time. Consequently, assessment must be an ongoing
process. It should not be assumed that a single learning experience
or assessment will provide a complete picture of students' intellectual
growth. The Evaluation Standards offer
many suggestions about this long-term assessment.
When interpreting these
standards, developing curriculum, and integrating evaluation procedures,
mathematics educators and others must realize that this broad, rich
curriculum is intended to be available to all students. No
student should be denied access to the study of one topic because
he or she has yet to master another.
The current curriculum
excludes many students from appreciating the useful, exciting, and
creative aspects of mathematics. The 5-8 standards outline a curriculum
that attempts to give all students the opportunity to appreciate
the full power and beauty of mathematics and acquire the mathematical
knowledge and intellectual tools necessary for its use in their
lives.
The chart on the next page
summarizes the major changes in emphasis for both the mathematical
content and instruction in grades 5-8.
SUMMARY OF CHANGES
IN CONTENT AND EMPHASIS IN 5--8 MATHEMATICS
INCREASED ATTENTION
PROBLEM SOLVING
- Pursuing open-ended problems
and extended problem-solving projects
- Investigating and formulating
questions from problem situations
- Representing situations
verbally, numerically, graphically, geometrically, or symbolically
COMMUNICATION
- Discussing, writing,
reading, and listening to mathematical ideas
REASONING
- Reasoning in spatial
contexts
- Reasoning with proportions
- Reasoning from graphs
- Reasoning inductively
and deductively
CONNECTIONS
- Connecting mathematics
to other subjects and to the world outside the classroom
- Connecting topics within
mathematics
- Applying mathematics
NUMBER/OPERATIONS/COMPUTATION
- Developing number sense
- Developing operation
sense
- Creating algorithms and
procedures
- Using estimation both
in solving problems and in checking the reasonableness of results
- Exploring relationships
among representations of, and operations on, whole numbers, fractions,
decimals, integers, and rational numbers
- Developing an understanding
of ratio, proportion, and percent
PATTERNS AND FUNCTIONS
- Identifying and using
functional relationships
- Developing and using
tables, graphs, and rules to describe situations
- Interpreting among different
mathematical representations
ALGEBRA
- Developing an understanding
of variables, expressions, and equations
- Using a variety of methods
to solve linear equations and informally investigate inequalities
and nonlinear equations
STATISTICS
- Using statistical methods
to describe, analyze, evaluate, and make decisions
PROBABILITY
- Creating experimental
and theoretical models of situations involving probabilities
GEOMETRY
- Developing an understanding
of geometric objects and relationships
- Using geometry in solving
problems
DECREASED ATTENTION
PROBLEM SOLVING
- Practicing routine, one-step
problems
- Practicing problems categorized
by types (e.g., coin problems, age problems)
COMMUNICATION
- Doing fill-in-the-blank
worksheets
- Answering questions that
require only yes, no, or a number as responses
REASONING
- Relying on outside authority
(teacher or an answer key)
CONNECTIONS
- Learning isolated topics
- Developing skills out
of context
NUMBER/OPERATIONS/COMPUTATION
- Memorizing rules and
algorithms
- Practicing tedious paper-and-pencil
computations
- Finding exact forms of
answers
- Memorizing procedures,
such as cross-multiplication, without understanding
- Practicing rounding numbers
out of context
PATTERNS AND FUNCTIONS
- Topics seldom in the
current curriculum
ALGEBRA
- Manipulating symbols
- Memorizing procedures
and drilling on equation solving
STATISTICS
PROBABILITY
GEOMETRY
- Memorizing geometric
vocabulary
- Memorizing facts and
relationships
SUMMARY OF CHANGES--continued
INCREASED ATTENTION
MEASUREMENT
- Estimating and using
measurement to solve problems
INSTRUCTIONAL PRACTICES
- Actively involving students
individually and in groups in exploring, conjecturing, analyzing,
and applying mathematics in both a mathematical and a real-world
context
- Using appropriate technology
for computation and exploration
- Using concrete materials
- Being a facilitator of
learning
- Assessing learning as
an integral part of instruction
DECREASED ATTENTION
MEASUREMENT
- Memorizing and manipulating
formulas
- Converting within and
between measurement systems
INSTRUCTIONAL PRACTICES
- Teaching computations
out of context
- Drilling on paper-and-pencil
algorithms
- Teaching topics in isolation
- Stressing memorization
- Being the dispenser of
knowledge
- Testing for the sole
purpose of assigning grades
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