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Model Academic Standards for Mathematics
Defining the Academic Standards
What are academic standards? Academic standards
specify what students should know
and be able to do, what they might be asked to do to give evidence of standards, and how
well they must perform. They include content, performance, and proficiency standards.
— Content standards refer to what students should know and
be able to do.
— Performance standards tell how students will show that they are meeting a standard.
— Proficiency standards indicate how well students must perform.
Why are academic standards necessary? Standards
serve as rigorous goals for teaching
and learning. Setting high standards enables students, parents, educators, and citizens to
know what students should have learned at a given point in time. The absence of standards
has consequences similar to lack of goals in any pursuit. Without clear goals, students may
be unmotivated and confused.
Contemporary society is placing immense academic demands
on students. Clear
statements about what students must know and be able to do are essential to ensure that
our schools offer students the opportunity to acquire the knowledge and skills necessary for
Why are state-level academic standards important?
Public education is a state
responsibility. The state superintendent and legislature must ensure that all children have
equal access to high quality educational programs. At a minimum, this requires clear
statements of what all children in the state should know and be able to do as well as
evidence that students are meeting these expectations. Furthermore, academic standards
form a sound basis on which to establish the content of a statewide assessment system.
Why does Wisconsin need its own academic standards?
Historically, the citizens of
Wisconsin are very serious and thoughtful about education. They expect and receive very
high performance from their schools. While educational needs may be similar among states,
values differ. Standards should reflect the collective values of the citizens and be tailored to
prepare young people for economic opportunities that exist in Wisconsin, the nation, and
Developing the Academic Standards
How were Wisconsin’s model academic standards
developed? Citizens throughout
the state developed the academic standards. The first phase involved educators, parents,
board of education members, and business and industry people who produced preliminary
content and performance standards in 12 subjects including English language arts,
mathematics, science, social studies, visual arts, music, theatre, dance, family and
consumer education, foreign language, health education, and physical education. These
standards are benchmarked to the end of grades 4, 8, and 12.
The next step required public input aimed at getting
information to revise and
improve the preliminary standards. This effort included forums and focus groups held
throughout the state. The state superintendent used extensive media exposure, including
telecommunications through the Department of Public Instruction (DPI) home page, to
ensure the widest possible awareness and participation in standards development.
Each subject had at least two drafts taken to the general
public for their review.
Based on this input, the standards were revised to reflect the values of Wisconsin’s citizens.
In January 1997, Governor Thompson appointed the
Governor’s Council on Model
Academic Standards. The Council augmented the existing Department of Public Instruction
task forces with additional appointees by the Council, these newly configured task forces
produced another draft of model academic standards for the subjects that are part of the
state assessment system. These include English language arts, mathematics, reading,
science, and social studies.
Once these draft standards were completed, public review
became the focus. Using a
series of statewide forums coupled with a wide mailing distribution and
telecommunications access through both the Wisconsin Department of Public Instruction
and the lieutenant governor’s home page, Wisconsin citizens provided valuable feedback. As
with previous drafts, all comments received serious consideration.
Who wrote the academic standards and what resources
were used? Each subject
area’s academic standards were drafted by teams of educators, parents, board of education
members, and business and industry people that were sub-groups of larger task forces. This
work was done after reviewing national standards in the subject area, standards from other
states, standards from local Wisconsin school districts, and standards like the nationwide
New Standards Project.
After the creation of the Governor’s Council on Model
Academic Standards, four task
forces representing English language arts (reading was folded into this group),
mathematics, science, and social studies were appointed. Combining the existing DPI task
force members with the Council’s appointees further ensured that the many voices of
Wisconsin’s citizenry were represented through the parents, educators, school board
members, and business and industry people sitting on those task forces. Documents
reviewed included the national standards in the subject area, standards from other states,
and standards from local Wisconsin schools. The two most frequently used resources were
the first and second drafts of content and performance standards developed by the DPI and
the Modern Red Schoolhouse standards developed by the Hudson Institute and Dr. Finley
How was the public involved in the standards process?
The DPI was involved in
extensive public engagement activities to gather citizen input on the first two drafts of the
academic standards. Over 19 focus group sessions, 17 community forums, and more than
450 presentations at conferences, conventions, and workshops were held. More than
500,000 paper copies of the standards tabloids were distributed across the state in addition
to more than 4,000 citizen visits to the standards on the DPI web page. Input from these
activities, along with more than 90 reviews by state and national organizations, provided
the writers with feedback on Wisconsin’s model academic standards.
Continuing the emphasis on public engagement started by
the DPI with previous
standards drafts, the Governor’s Council on Model Academic Standards held nine
community forums. In addition, more than 50,000 paper copies of the standards were
distributed. Recipients included each public school building as well as all major education
stakeholders and interest groups. Lending the prestige of their offices to the standards
development, members of the Council met with editorial boards of media outlets throughout
the state discussing the model academic standards.
Will academic standards be developed in areas other
than the 12 areas listed
above? Yes, currently the DPI has convened six task forces to develop academic standards
in agriculture, business, environmental education, marketing, technology education, and
information and technology literacy. Task force members include educators, parents, school
board members, and representatives of business and industry. These academic standards
will be completed by the start of the 1998-99 school year.
Using the Academic Standards
How will the Department of Public Instruction use the
Academic Standards? Upon completing its work, the Governor’s Council on Model
Academic Standards submitted academic content and performance standards for English
language arts, mathematics, science, and social studies to the governor. On January 13,
1998, Governor Thompson signed Executive Order 326, thus approving and issuing the
model academic and performance standards developed by the Governor’s Council. These
approved standards will be used as the basis for state testing, especially as it relates to the
Wisconsin Reading Comprehension Test, the Wisconsin Student Assessment System, and
the planned High School Graduation Test.
Additionally, the DPI will use the Wisconsin Model
Academic Standards as the basis
for revision of its Guides to Curriculum Planning and as the foundation for professional
development activities that it sponsors.
Must a district adopt the Wisconsin Model Academic
Standards? Adopting the
Wisconsin Model Academic Standards is voluntary, not mandatory. Districts, however,
must have academic standards in place by August 1, 1998. At a minimum, districts are
required to have standards in reading and writing, geography and history, mathematics,
and science. Districts may adopt the model state standards, or standards from other
sources, or develop their own standards.
How will local districts use the academic standards?
Districts may use the academic
standards as guides for developing local grade-by-grade level curriculum. Implementing
standards may require some school districts to upgrade school and district curriculums. In
some cases, this may result in significant changes in instructional methods and materials,
local assessments, and professional development opportunities for the teaching and
What is the difference between academic standards and
curriculum? Standards are
statements about what students should know and be able to do, what they might be asked
to do to give evidence of learning, and how well they should be expected to know or do it.
Curriculum is the program devised by local school districts used to prepare students to
meet standards. It consists of activities and lessons at each grade level, instructional
materials, and various instructional techniques. In short, standards define what is to be
learned at certain points in time, and from a broad perspective, what performances will be
accepted as evidence that the learning has occurred. Curriculum specifies the details of the
day-to-day schooling at the local level.
What is the link between statewide academic standards
and statewide testing?
Statewide academic standards in mathematics, English language arts, science, and social
studies determine the scope of statewide testing. While these standards are much broader
in content than any single Wisconsin Student Assessment System (WSAS) test, they do
describe the range of knowledge and skills that may appear on the tests. If content does not
appear in the academic standards, it will not be part of a WSAS test. The statewide
standards clarify what must be studied to prepare for WSAS tests. If students have learned
all of the material indicated by the standards in the assessed content areas, they should do
very well on the state tests.
Relating the Academic Standards to All Students
Parents and educators of students with disabilities, with
limited English proficiency
(LEP), and with accelerated needs may ask why academic standards are important for their
students. Academic standards serve as a valuable basis for establishing meaningful goals
as part of each student’s developmental progress and demonstration of proficiency. The
clarity of academic standards provides meaningful, concrete goals for the achievement of
students with exceptional education needs (EEN), LEP, and accelerated needs consistent
with all other students.
Academic standards may serve as the foundation for
decisions for students with EEN, LEP, and accelerated needs. While the vast majority of
students with EEN and LEP should be expected to work toward and achieve these
standards, accommodations and modifications to help these students reach the achievement
goals will need to be individually identified and implemented. For students with EEN,
these decisions are made as part of their individualized education program (IEP) plans.
Accelerated students may achieve well beyond the academic standards and move into
advanced grade levels or into advanced coursework.
Clearly, these academic standards are for all students. As
our state assessments are
aligned with these standards and school districts adopt, adapt, or develop their own
standards and multiple measures for determining proficiencies of students, greater
accountability for the progress of all students can be assured. In Wisconsin this means all
students reaching their full individual potential, every school being accountable, every
parent a welcomed partner, every community supportive, and no excuses.
Applying the Academic Standards Across the Curriculum
When community members and employers consider what they
want citizens and
employees to know and be able to do, they often speak of broad areas of applied knowledge
such as communication, thinking, problem solving, and decision making. These areas
connect or go beyond the mastery of individual subject areas. As students apply their
knowledge both within and across the various curricular areas, they develop the concepts
and complex thinking of educated persons.
Community members need these skills to function as
Employers prize those employees who demonstrate these skills because they are people who
can continue learning and connect what they have learned to the requirements of a job.
College and university faculty recognize the need for these skills as the means of developing
the level of understanding that separates the expert from the beginner.
Teachers in every class should expect and encourage the
development of these
shared applications, both to promote the learning of the subject content and to extend
learning across the curriculum. These applications fall into five general categories:
1) Application of the Basics
2) Ability to Think
— Problem solving
— Informed decision making
— Systems thinking
— Critical, creative, and analytical
— Imagining places, times , and
situations different from one ’s
— Developing and testing a
— Transferring learning to new
3) Skill in Communication
— Constructing and defending an
— Working effectively in groups
— Communicating plans and
processes for reaching goals
— Receiving and acting on
instructions, plans, and models
— Communicating with a variety of
tools and skills
4) Production of Quality Work
— Acquiring and using information
— Creating quality products and
— Revising products and
— Developing and pursuing positive
5) Connections with Community
— Recognizing and acting on
responsibilities as a citizen
— Preparing for work and lifelong
— Contributing to the aesthetic and
cultural life of the community
— Seeing oneself and one’s
community within the state,
nation, and world
— Contributing and adapting to
scientific and technological
Overview of Mathematics
The Wisconsin Model Academic Standards for Mathematics
have been developed by a team
of Wisconsin citizens including classroom teachers, professional educators, parents,
business persons, and school board members. The team used various resource documents in
its deliberations, including the Curriculum and Evaluation Standards for School
Mathematics of the National Council of Teachers of Mathematics (NCTM), A Guide to
Curriculum Planning in Mathematics of the Wisconsin Department of Public Instruction
(DPI), Modern Red Schoolhouse, and state standards and guidelines developed by other
entities such as the New Standards Project, the National Institute for Educational
Research in Japan, and the states of California, Colorado, Michigan, Oregon, and Virginia.
The Wisconsin Mathematics Standards are designed to be
general guidelines which may be
adopted or adapted by local school districts with cooperation and input from parents and
other concerned citizens. They are not meant to be a full curriculum nor a prescription for
instructional practice carried out from week-to-week in classrooms. They are important
goals for ALL students from which individual schools can build a complete curriculum
specific to their district’s and children’s needs.
The content of these standards reflects the shift in
mathematical emphasis necessitated by
technological advances in an information society. Some topics in mathematics (i.e.,
estimation, place value) have become more important and several new areas in
mathematics (i.e., fractals, predictive statistics) have developed. The understanding of
mathematical concepts has become imperative for each citizen as everyday functions
become more mathematically complex and as low-skill jobs become nonexistent.
Mathematics instruction must, therefore, be made
accessible, understandable, and
meaningful for all students. Access includes:
• learning experiences that enable students to acquire and
build knowledge and skills
• different instructor roles that use a variety of teaching techniques, adapting them as
appropriate for different purposes of instruction and students’ needs
• adaptive learning environments so that all students achieve success, and
• servicing all students, including populations previously sometimes underserved, i.e.,
EEN, LEP, etc.
Likewise, mathematics assessment must address the
understanding of all students (as
assessed with accommodations which match instructional methodologies ) so that
mathematics instruction can be evaluated and improved for all students.
Not all students in secondary school elect to pursue
“college preparatory” courses.
Therefore, the following content and performance standards do not reflect the content of
those higher level courses. Rather, they reflect the content of a core mathematical
experience that ALL students should encounter, regardless of what mathematics courses
These standards are intended as points of reference, not
limitations. Some students will
accomplish much more than these standards envision; yet the standards set the targets for
what all students should be challenged, encouraged, and expected to achieve.
Goals and Instructional Practice
Classroom practice geared to the attainment of the
Wisconsin standards should be aimed at
creating a community of learners and scholars, a place where the teachers and students
actively investigate and discuss mathematical ideas, using a wide variety of tools,
materials, and technology. Classes should engage students in more high-level mathematical
thought and emphasize conceptual understanding, more so than in the past.
Important goals for students are:
• to develop a deep conceptual understanding in order to
make sense of mathematics
(students need to know not only how to apply skills and knowledge, but also when to apply
them and why they are being applied)
• to master specific knowledge necessary for its application to real problems , for the study
of related subject matter, and for continued study in mathematics
• to learn and view mathematics as a way of thinking about and interpreting the world
• to recognize that mathematics is a creative part of human culture in much the same
way as the fine arts
Mathematics should be viewed as a unified whole made up of
connected, big ideas rather
than as a disjointed collection of meaningless, abstract ideas and skills. Learning is easier
when students see the connections between various concepts and procedures, and between
the various branches of mathematics. Students should also be aware of the connections
between, and applications of, mathematics and other disciplines, such as the sciences, art,
music, business, medicine, and government.
Mathematics is important because its concepts and
procedures can be applied to the
solution of problems of varying kinds and complexity. Solving problems challenges students
to apply their conceptual understanding in a new or complex situation, to exercise their
basic skills, and to see mathematics as a way of finding answers to many of the problems
they encounter both within and outside the classroom. Students grow in their ability and
persistence in problem solving through extensive classroom experience in posing,
formulating, and solving problems at a variety of levels of difficulty and at every level in
their mathematical development.
The ability to reason is such a vital part of mathematical
behavior that it is safe to assert
that mathematics cannot be done without it. At all levels, students should be able to
provide a reason why they have chosen to apply a particular skill or concept, or why that
skill works the way it does. Further, students should habitually check their results and
conclusions for their reasonableness; that is, “does this make sense?” Proportional and
spatial reasoning are specific kinds of reasoning that all students should have at their
disposal. And, finally, it is important that all students should be able to apply the logical
reasoning skills of induction and deduction to make, test, and evaluate mathematical
conjectures, to justify steps in mathematical procedures, and determine whether
conclusions are valid by analyzing an argument.
Whether working alone, or as part of a team, students must
be able to communicate their
thinking to others. Students must learn not only the signs, symbols, and specialized terms
of mathematics, but also how to use this mathematical language in oral, symbolic, and
written communication. These communication skills become even more relevant when
students leave their classroom world for the world of work.
Calculators , computers, spreadsheets, graphing utilities,
and other forms of electronic
information technology are now standard tools for mathematical problem solving in science,
engineering, business, medicine, government, and finance. Thus, the use of technology must
be an integral part of teaching and learning mathematics. Such use should aim at
enhancing conceptual understanding and problem-solving skills. However, the tools of
technology are not a substitute for proficiency in basic computational skills.
In the text that follows, terms with an asterisk (*) are
defined and/or exemplified in the
Glossary of Terms following which appears on page 22 of this document.