Mathematics For All
Pre-Kindergarten
Through Grade 12
Standards
Developed for Massachusetts Students
by Massachusetts Educators
A Work in Progress
Mathematics For All
November 2, 2000
Page 2 of 55
Why Produce Mathematics For All?
The effort to determine preK-12 learning standards for mathematics in the Commonwealth of Massachusetts
has been an arduous journey over the past eight years. With great hope, many individuals have worked with
our state’s policy makers to consider, recommend and write portions of Mathematics Frameworks during this
period. Under the umbrella of Education Reform, educators have written standards and expectations, and
also helped to create the Massachusetts Comprehensive Assessment System. All of this has been done with
the singular goal of providing a strong foundation in mathematics for all of the Commonwealth’s children.
During the past year, this process has broken down to the point where the recently approved, revised version
of a Mathematics Curriculum Framework no longer represents the viewpoint of mathematics education
shared by previous writers or, we believe, the philosophy of the National Council of Teachers of
Mathematics (NCTM) as espoused in the April 2000 release of Principles and Standards for School
Mathematics. In Mathematics For All, educators from across the Commonwealth offer an alternative set of
broad learning standards that, we are fully aligned with NCTM and also reflect curriculum decisions that
have been made in Massachusetts’ school districts over the past decade.
We urge our colleagues in Massachusetts to evaluate Mathematics For All, both as a guide to how we
educate our children and in order to continue the dialogue to create an exemplary system of mathematics
education. A common voice is needed to prevent the production of another generation of mathematics
phobics as has been the legacy of times past.
Key features of Mathematics For All:
 There is a common set of Learning Standards and Learning Expectations for all children.
 The focus is on teaching for understanding.
 The Learning Standards are those of NCTM’s Principles and Standards for School Mathematics.
 The breadth within these Learning Standards ensures rich meaning for students of all abilities.
 The Learning Standards are not a list of what is to be tested through a pencil-and-paper format;
rather, a variety of assessments should be used to evaluate student learning.
 The Learning Expectations include explorations of concepts that are mastered in later years.
We need the critical feedback of our colleagues in the Massachusetts mathematics education community to
strengthen this document and to make it the one reference representing the best thinking of our profession.
We encourage and welcome your comments, criticisms and suggestions. Currently, the most effective
method to provide feedback is to visit the Massachusetts Educators for Mathematical Excellence (MEME)
discussion site on the web at http://mathforum.com/discuss/meme. In future months, writers of Mathematics
For All will reach out in a variety of forums that will be advertised on the MEME discussion site or through
other means. [We are very grateful to Helen Plotkin and Richard Tchen, creators of MathForum.com and
WebCT.com, the e-Learning Hub, who have offered space on their website for Massachusetts educators.]
Points to consider:
 What are the strengths of this document?
 What areas need improvement, and what suggestions do you make?
 Would you be willing to work on revisions?
 Would you like your name to be included as a contributor in the final document?
 Please provide your name, affiliation and e-mail address in the MEME discussion.
Mathematics For All is still very much a “work in progress.” Further editions of Mathematics For All will
evolve as we work together to make this the document guiding mathematics instruction in Massachusetts.
Mathematics For All
November 2, 2000
Page 3 of 55
Table of Contents
Page Numbers
A Message to Readers of Mathematics For All
Table of Contents
Acknowledgements
Preface and History of Development
Organization of Mathematics For All
Guiding Philosophy
Habits of Mind
Guiding Principles
Strand Overviews
Strand 1: Number Sense and Operations
Strand 2: Patterns, Functions and Algebra
Strand 3: Geometry
Strand 4: Data Analysis, Statistics and Probability
Strand 5: Measurement
Standards and Learning Expectations
PreK-K
Grades 1-2
Grades 3-4
Grades 5-6
Grades 7-8
Grades 9-10
Grades 11-12
Appendix A: Criteria for Evaluating Instructional Materials and Programs in Mathematics
2
3
4
5-6
7
8-10
11
12-20
21-23
24
25
26
27
28-30
31-34
35-39
40-43
44-48
49-51
52
53-55
Many contributors to Mathematics for All have worked closely with the Massachusetts Department of
Education creating both the 1996 and the 2000 editions of the Mathematics Curriculum Frameworks.
Therefore, similar text occasionally appears in Mathematics for All and the Department’s documents.
Mathematics For All
November 2, 2000
Page 4 of 55
Acknowledgements
Mathematics For All was developed by a group of teachers and administrators, mathematicians, university
mathematics educators and other community members. Many more people lent their support over the past
nine months of work, but we did not want to include their names without their permission. We hope that our
list will grow exponentially as Mathematics For All is distributed and critiqued by many more Massachusetts
mathematics educators.
Endorsement by Massachusetts’ Mathematics Organizations for Teachers
Dwayne Cameron, President: Association of Teachers of Mathematics in Massachusetts
Claire Zalewski Graham, President: Boston Area Mathematics Specialists
Spike Clancy, President: Association of Teachers of Mathematics in Western Massachusetts
Lyn Heady, President-Elect: Association of Teachers of Mathematics in Western Massachusetts
Ruth O'Malley, President: Association of Teachers of Mathematics in New England
Leadership Endorsement from Past DOE Statewide Mathematics Coordinators
Anne Collins, Boston College (1997-1999)
Peg Bondorew, CESAME, Northeastern University (1993-1995)
Gisele Zangari, Boston University Academy (1995-1997)
Contributors Who Have Worked on Mathematics For All
Jane Albert, Concord Public Schools
Sheldon Berman, Hudson Public Schools
Peg Bondorew, Northeastern University
John Bookston, Boston Public Schools
Nancy Buell, Brookline Public Schools
Michael Bresnahan, Cambridge Public Schools
Arthur Camins, Hudson Public Schools
Ricky Carter, ARC Implementation Center
Maureen Chapman-Fahey, Medford Public Schools
Rose Christiansen, Brookline Public Schools
Spike Clancy, Ludlow Public Schools
Anne Collins, Boston College
Mary Eich, Newton Public Schools
Rebeka Eston, Lincoln Public Schools
Patricia C. Foley, Westborough Public Schools
Thomas E. Foley, Waltham Public Schools
Christine Francis, Concord Public Schools
Maurice Gilmore, Northeastern University
Claire Zalewski Graham, Framingham State College
Carole Greenes, Boston University
Claire Groden, Watertown Public Schools
Barbara Haig, Northborough Public Schools
Maggi Hartnett, Ayer Public Schools
Deborah Hughes Hallett, University of Arizona
James Hamos, Univ. of Massachusetts Medical School
Mary Hogan, Boston College
Neelia Jackson, Boston Public Schools
James Kaput, University of Massachusetts-Dartmouth
Bill Kendall, Braintree Public Schools
Margaret Kenney, Boston College
Mary Jo Livingstone, Weymouth High School
Christopher Martes, MASS
Cliff Martin, Whitman Hanson High School
Joan Martin, Newton Public Schools
William Masalski, Univ. of Massachusetts-Amherst
Nancy McLaughlin, Lawrence Public Schools
Jan Mokros, TERC
Gloria Moran, Bridgewater-Raynham Public Schools
Christine Moynihan, Wayland Public Schools
Blake Munro, Wellesley Public Schools
Nancy Nichols, Saugus Public Schools
Ruth O'Malley, ATMNE
Margaret Riddle, Northampton Public Schools
Jan Rook, Boston Public Schools
Leanna Russell, E. Bridgewater High School
Mary Sapienza, Newton North High School
Paula Sennett, Silver Lake Regional Public School
Debra Shein-Gerson, Brookline Public Schools
Victor Steinbok, Boston University
J. Bryan Sullivan, Hudson Public Schools
Karen Tripoli, Lexington Public Schools
Rhonda Weinstein, Brookline Public Schools
Susan Weiss, Solomon Schechter Day School, Newton
Carolyn Wyatt, Newton Public Schools
We wish to acknowledge the consistent and substantive support of our friends and colleagues at the Massachusetts
Teachers Association, particularly Laura Barrett, Ralph Devlin, and Kathleen Skinner. Their encouragement has kept
us going at critical junctures in our process, and their knowledge and resources have been crucial as we move towards
truly making Mathematics for All a key document for teachers across the Commonwealth.
Mathematics For All
November 2, 2000
Page 5 of 55
Preface and History of Development
Mathematics For All is a framework for mathematics curriculum that was developed by a group of teachers,
administrators, mathematicians, university mathematics educators, and other community members who worked
under the leadership of a Steering Committee of representatives of the Massachusetts Advisory Council for
Mathematics and Science, the Massachusetts Association of School Superintendents (M.A.S.S.), the
Association of Teachers of Mathematics in Massachusetts (ATMIM), and the Association of Teachers of
Mathematics in Western Massachusetts (MATHWEST). This committee was appointed by the Commissioner
of Education in February, 2000, and was given the task of providing recommendations for the draft
Massachusetts Mathematics Framework that would ensure mathematical appropriateness and align the draft
document with the new National Council of Teachers of Mathematics Principles and Standards for School
Mathematics, released in April, 2000. While a revised Mathematics Framework was approved by the Board of
Education in July 2000, those of us who have worked with the Department continue to believe that there is a
need for a standards document that reflects Principles and Standards for School Mathematics, a national
endeavor based on current research in mathematics education. Mathematics For All is based on this as well as
other documents that share the vision that all students must have access to high quality mathematics programs
that support successful learning of mathematics and help them develop a mathematical sense and intuitive
understanding. This document also continues to support the Policy statement on Mathematics and Science
Education, adopted by the Massachusetts Board of Education in 1992: Mathematics and science as academic
disciplines and tools for problem solving are central to the vitality of the economy and quality of life. They
offer students of all ages opportunities to embark on adventures that stimulate the intellect and imagination.
The first Massachusetts Mathematics Framework, Achieving Mathematical Power, adopted in June 1996, was
based upon two reform initiatives in Massachusetts, the Education Reform Act of 1993 and Partnerships
Advancing the Learning of Mathematics and Science (PALMS). PALMS is the Statewide Systemic Initiative,
a collaborative effort jointly funded by the National Science Foundation and the Commonwealth of
Massachusetts, which began in 1992. Of the seven initial goals for this initiative, the first was to develop,
disseminate, and implement curriculum frameworks in Mathematics and Science & Technology. With the
passage of the Massachusetts Education Reform Act in June 1993 and additional funding from the U.S.
Department of Education, development of the curriculum frameworks was extended to include grades 9-12 and
Adult Basic Education.
The creation of Massachusetts’ first mathematics framework was a collaborative endeavor among members of
the Framework Development Committee--teachers, school and district administrators, mathematicians, college
faculty, parents, and representatives of business and community organizations across the state. A majority of
the members were classroom teachers with extensive experience teaching mathematics at elementary, middle,
and high school levels. Committee meetings were convened to consider each draft framework from the
standpoints of clarity, accessibility, consistency, pedagogy, mathematical correctness, and alignment with the
Massachusetts Common Core of Learning and the National Council of Teachers of Mathematics Curriculum
and Evaluation Standards.
The core concept of Achieving Mathematical Power was that students develop mathematical power through
problem solving, communication, reasoning and connections. The mathematics framework was more than a
collection of concepts and skills. For each individual it involved methods of investigating and reasoning, means
of communication, notions of context, and development of personal self-confidence. The framework provided
quality and equity for all learners. The Guiding Principles and Habits of Mind of this framework outlined ways
in which this could become a reality.
The Mathematics Content section presented an outline upon which district and school curricula, instruction, and
assessment could be developed. Examples of student learning, vignettes, models, diagrams, and graphics
contextualized and enhanced the standards. The goals for all learners in the Massachusetts mathematics
Mathematics For All
November 2, 2000
Page 6 of 55
framework were that they value mathematics, become confident in their ability to know and to do
mathematics, become mathematical problem solvers, and learn to reason and communicate
mathematically. The document presented here, Mathematics For All, is an attempt to build on the strengths of
Achieving Mathematical Power and continue the Vision espoused by PALMS:
"All Massachusetts students will receive a high quality, hands-on education in mathematics and in
science & technology that empowers them to be productive, problem-solving citizens and workers.
Partnerships among businesses, institutions of higher education, policy makers, governmental
agencies, cultural institutions, teachers and families will create a rich learning environment and
provide a lasting foundation for continual improvement.
Challenging standards for content, teaching methods and equity defined in statewide curriculum
frameworks will guide district practice. Learning will be active, built on discovery and reflection and
a variety of assessments will test for understanding. New teachers will enter the profession with a
solid grounding in mathematics and science content and in effective strategies for engaging a diversity
of learners. Experienced teachers will continually deepen their knowledge and professional skills.
PALMS will be the vanguard of education reform in Massachusetts." (www.doe.mass.edu/palms)
In November 1998, the Department of Education convened a Revision Committee of mathematics teachers,
mathematicians, and university mathematics educators to examine Achieving Mathematical Power and to make
recommendations that would provide additional guidance to the school districts. Their recommendations
included more specificity for the content standards in terms of grade level spans, as well as clarification of
many of the original standards, to help guide the ongoing development of the Massachusetts Comprehensive
Assessment System. The standards developed by the Revision Committee formed the basis for the additional
work of the group of teachers and administrators, mathematicians, university mathematics professors, and other
community members who have developed Mathematics For All.
In April 2000, the National Council of Teachers of Mathematics released the Principles and Standards for
School Mathematics,1 a document that builds on and extends the original NCTM Standards documents.
Educational research served as a basis for many of the proposals and claims made in the Principles and
Standards about what is possible for students to learn about certain content areas at certain levels and under
certain pedagogical conditions. The philosophy in Mathematics For All is directly aligned with that in the
Principles and Standards, as well as the Vision espoused by PALMS. Indeed, the writers of Mathematics For
All believe that Massachusetts’ standards should be the NCTM Learning Standards in the Principles and
Standards for School Mathematics. All these efforts call for a common foundation of mathematics to be
learned by all students. Mathematics For All is the result of educators’ zeal to provide that foundation.
1
National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Reston, VA: 2000.
Throughout Mathematics For All, this recent NCTM document is cited as NCTM 2000.
Mathematics For All
November 2, 2000
Page 7 of 55
Organization of Mathematics For All
The document is separated into four sections: Guiding Philosophy, Habits of Mind, Guiding Principles,
Content Strands, and an Appendix. The Guiding Philosophy provides the Vision for Mathematics For All,
and includes the Core Concept of the framework. The components of the Core Concept are Problem Solving,
Communicating, Reasoning and Proof, Making Connections and Representation. The NCTM Process
Standards are included with each of these components. The Habits of Mind describe desirable attitudes
toward learning and using mathematics. The Guiding Principles articulate a set of beliefs about teaching,
learning, and assessing mathematics. The Learning Standards are organized by grade span and specify the
understanding, knowledge and skills that all students should acquire. The grade spans are: preK-K, 1-2, 3-4, 56, 7-8, and 9-10. Within each grade span the Learning Standards are grouped into Strands. The strands are
Number Sense and Operations; Patterns, Functions, and Algebra; Geometry; Data Analysis, Statistics, and
Probability; and Measurement. The learning standards are further organized into two components, NCTM
Learning Standards and Learning Expectations. The NCTM Learning Standards, from the Principles and
Standards for School Mathematics, designate the Curriculum Standards for each grade span. The Learning
Expectations provide additional specificity and can be used to inform the development of assessment tasks.
The Appendix is an evaluation rubric that can be used by decision-makers in districts to appraise the degree to
which instructional materials and programs match the philosophy, principles and learning expectations
embedded within Mathematics For All. The Criteria for Evaluating Instructional Materials and Programs in
Mathematics was adapted from an Evaluation Tool found on the CESAME (Center for the Enhancement of
Science and Math Education, Northeastern University) web site – www.cesame.neu.edu – for the IMPACT
Project.
The organization of the standards into courses at the secondary level is at the discretion of the local school
districts. Indeed, the writers of Mathematics For All have struggled with identifying Learning Standards for
grades 11-12, and look forward to rich discussions with colleagues across the Commonwealth to identify late
high school standards that align with expectations from higher education and/or the world of work.
According to the Principles and Standards for School Mathematics, "A school mathematics curriculum should
provide a road map that helps teachers guide students to increasing levels of sophistication and depths of
knowledge. …A well-articulated curriculum gives teachers guidance regarding important ideas or major
themes, ...It also gives guidance about the depth of study warranted at particular times and when closure is
expected for particular skills or concepts." (NCTM 2000, p. 16) Throughout this document, the standards are
written to allow time for study of additional challenging material at every grade level and for advanced courses
in high school. All schools should provide further work in mathematics through advanced placement courses,
independent research, internships, or study of special topics.
Mathematics For All
November 2, 2000
Page 8 of 55
Guiding Philosophy
The vision of Mathematics For All is of students acquiring an understanding of fundamental mathematical
concepts that will enhance their lives. It is “one in which students engage in purposeful activities that grow out
of problem situations, requiring reasoning and creative thinking, gathering and applying information,
discovering, inventing, and communicating ideas, and testing those ideas through critical reflection and
argumentation.” (Thompson, 1992, p. 128)2 Students will have necessary skills to succeed in the workplace,
solve practical problems around their homes, understand statistics and use their analytical abilities. Acquiring
such an understanding depends on many factors. In particular, it depends on a clear, comprehensive, coherent
and developmentally appropriate set of standards. It also depends on students learning to think in mathematical
ways. Mathematics For All envisions that all students in the Commonwealth will acquire an
understanding of fundamental mathematical concepts (i.e., the Learning Standards and Learning
Expectations) through a strong mathematics program that emphasizes problem solving, communicating,
reasoning and proof, making connections and using representations.
Problem Solving
Problem solving is a powerful means of developing students’ knowledge of mathematics and an indispensable
outcome of a good mathematics education, and as such, it is an essential component of the curriculum. A
mathematical problem, as distinct from an exercise, requires the solver to determine a solution method.
Therefore, mathematical problem solving requires understanding concepts, procedures, and strategies. To
become good problem solvers, students need many opportunities to formulate questions, model problem
situations using a variety of means, generalize mathematical relationships, and solve problems in both
mathematical and real-world contexts. "Solving problems is not only a goal of learning mathematics but also a
major means of doing so." (NCTM 2000, p. 52) Developing and applying strategies to a wide variety of
problems can help students develop a strong command of mathematics content.
"Instructional programs from pre-Kindergarten through grade 12 should enable all students to –
I.
build new mathematical knowledge through problem solving;
II.
solve problems that arise in mathematics and in other contexts;
III.
apply and adapt a variety of appropriate strategies to solve problems;
IV.
monitor and reflect on the process of mathematical problem solving." (NCTM 2000, p. 52).
Communicating
The ability to express mathematical ideas coherently to different audiences is an important skill in a
technological society. Students develop this skill and deepen their understanding of mathematics as they use
accurate mathematical language to talk and write about what they are doing. They clarify mathematical ideas
and definitions as they collaborate with other students, discuss with peers and experts, and reflect on and share
ideas, strategies, and solutions. Reading in the content area of mathematics helps students understand and
develop the skills of making convincing arguments and representing mathematical ideas verbally, pictorially,
Thompson, A.G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.),
Handbook of research on mathematics teaching and learning (pp. 127-146). New York: Macmillan.
2
Mathematics For All
November 2, 2000
Page 9 of 55
and symbolically. Reading what students write and listening carefully to what students say are excellent ways
for teachers to identify students’ understandings and misconceptions.
"Instructional programs from pre-Kindergarten through grade 12 should enable all students to -I.
organize and consolidate their mathematical thinking through communication;
II.
communicate their mathematical thinking coherently and clearly to peers, teachers, and others;
III.
analyze and evaluate the mathematical thinking and strategies of others;
IV.
use the language of mathematics to express mathematical ideas precisely" (NCTM 2000, p. 60).
Reasoning and Proof
From the early grades, students develop their reasoning skills by making and testing mathematical conjectures,
drawing logical conclusions, and justifying their thinking in developmentally appropriate ways. As they
advance through the grades, students’ arguments become more sophisticated and they are able to construct
formal proofs. "Reasoning mathematically is a habit of mind, it must be developed through consistent use in
many contexts." (NCTM 2000, p. 56) With multiple opportunities to use patterns in mathematical problems,
analyze mathematical situations, and deduce conclusions, students learn what mathematical reasoning entails.
"Instructional programs from pre-Kindergarten through grade 12 should enable all students to -I.
Recognize reasoning and proof as fundamental aspects of mathematics;
II.
make and investigate mathematical conjectures;
III.
develop and evaluate mathematical arguments and proofs;
IV.
select and use various types of reasoning and methods of proof." (NCTM 2000, p. 56).
Making Connections
"Mathematics is not a collection of separate strands or standards, even though it is often partitioned and
presented in this manner. Rather, mathematics is an integrated field of study," (NCTM 2000, p. 64) Students
develop a perspective of the mathematics field as an integrated whole by understanding connections within and
outside of the discipline. It is important for teachers to demonstrate the significance and relevance of the
subject by exploring the connections that exist within mathematics, with other disciplines, and between
mathematics and students’ own experiences.
"Instructional programs from pre-Kindergarten through grade 12 should enable all students to -I.
recognize and use connections among the mathematical ideas;
II.
understand how mathematical ideas interconnect and build on one another to produce a coherent
whole;
III.
recognize and apply mathematics in contexts outside of mathematics." (NCTM 2000, p. 64).
Mathematics For All
November 2, 2000
Page 10 of 55
Representations
Mathematical representations come in many forms. They can be numerals or diagrams, they can be algebraic
expressions or graphs, and they can be matrices that model a method for solving a system of equations. "When
students gain access to mathematical representations and the ideas they represent, they have a set of tools that
significantly expand their capacity to think mathematically." (NCTM 2000, p. 67) Students gain this access as
they develop strategies for diagramming, analyzing and solving problems. "The importance of using multiple
representations should be emphasized throughout students’ mathematical education." (NCTM 2000, p. 69)
"Instructional programs from pre-Kindergarten through grade 12 should enable all students to -I.
create and use representations to organize, record, and communicate mathematical ideas;
II.
select, apply and translate among mathematical representations to solve problems;
III.
use representations to model and interpret physical, social, and mathematical phenomena." (NCTM
2000, p. 67).
Mathematics For All
November 2, 2000
Page 11 of 55
Habits of Mind
Developing mathematical competence includes the gradual acquisition of ways of thinking and behaving called
Mathematical Habits of Mind. These habits might be thought of as desirable attitudes toward learning and
using mathematics that should become an integral part of each student's approach to mathematics. They derive
from the curriculum and are reinforced when teachers model these habits for students.
The Mathematics Habits of Mind are listed here in the form of questions to be considered and modeled by
teachers and shared with students. They help frame the vision of achieving mathematical competence through
Problem Solving, Reasoning and Proof, Communication, Connections, and Representation, and thus are
integral to achieving Mathematics For All.
Mathematical Habits of Mind: A Guide for Reflection

How do I use my mathematical skills to interpret information and solve problems? Mathematics can
provide a venue for us to work beyond our center of competence, encouraging us to push the limits of
our mathematical knowledge and abilities.

Do I communicate to others how I solved a problem or justified my solution? Seeking feedback helps
us to deepen our mathematical understanding by reflecting upon, extending and refining our thinking.

In what ways do I reflect confidence in my ability to do mathematics? Confidence in mathematical
ability brings with it an attitude of persistence when solutions are not apparent.

In what ways do I explore the relationship of mathematics to other areas that interest me? To other
subject areas? When we explore mathematics in the course of our daily lives, we encourage the use of
available resources as well as integrate mathematics within our existing network of ideas.

How do I show that I value and appreciate the beauty and fascination of mathematics? Valuing all
dimensions of mathematics encourages us to view mathematics in unconventional ways, generating
new ways of thinking.
Mathematics For All
November 2, 2000
Page 12 of 55
Guiding Principles
These are the underlying beliefs and tenets central to the vision of mathematical process and content standards
for mathematics education in Massachusetts.
Guiding Principle I: Equity - All students have access to, and support to succeed in, high quality mathematics
programs.
Guiding Principle II: Curriculum - Mathematics curriculum should be coherent and focused on important
mathematics and well articulated across the grades.
Guiding Principle III: Learning - Students explore mathematical ideas in ways that maintain their enjoyment of
and curiosity about mathematics, help them develop depth of understanding, and reflect real-world
applications.
Guiding Principle IV: Teaching - Mathematics instruction involves understanding what students know and
need to learn and then engaging them to learn it well.
Guiding Principle V: Assessment - Mathematics assessment is a multifaceted tool that monitors student
performance, improves instruction, enhances learning, and encourages student self-reflection
Guiding Principle VI: Technology - Technology is an essential tool for effective mathematics education.
Guiding Principle VII: Lifelong Learning - Mathematics learning is a lifelong process that begins and
continues in the home and extends to school and community settings.
Mathematics For All
November 2, 2000
Page 13 of 55
Guiding Principle I: Equity
All students must have access to, and support to succeed in, high quality mathematics programs.
The underrepresentation of certain groups in mathematics, science, and technology education is
well documented. The Statewide Systemic Initiative seeks to effect change in student
participation by transforming school curriculum, classroom instruction, and teacher education,
and by increasing parental support so that all students have opportunities suited to their needs to
learn mathematics, science, and technology. -- National Science Foundation, Equity Framework
in Mathematics, Science, and Technology Education
"All students should have access to an excellent and equitable mathematics program that provides solid support
for their learning and is responsive to their prior knowledge, intellectual strengths, and personal interests." To
promote achievement of these standards, teachers should encourage classroom talk, reflection, use of multiple
problem-solving strategies, and a positive disposition toward mathematics. They should have high expectations
for all students. (NCTM 2000, pp. 12-13)
All students must have access to high quality mathematics programs that support successful learning of
mathematics and help them to develop a mathematical sense and intuitive understanding. It is our
responsibility to ensure that students are fairly represented in each mathematics program and have equitable
access to resources. Everyone learns best in an environment that acknowledges, respects, and accommodates
each learner's background, learning style, and gender. For example, a teacher's listening attentively to all
students' ideas helps to foster in students a sense of control of their future. Or, by making special efforts to
achieve classroom integration when students self-segregate, a teacher enhances students' respect for others'
backgrounds and learning styles.
All students should see themselves as mathematicians, capable of using their evolving mathematical
competence to solve new problems. Some students may ultimately progress further in their mathematical
learning than others, and learning may take different amounts of time for different learners. However, if each
student is offered an accessible approach to learning mathematics, consistent with his learning style and
experience, then all students can learn mathematics. This means establishing high standards of expectations
and helping students when they are struggling with mathematics. It is not enough to enroll students in higher
level classes; everything possible must be done to engage their interests. Each member of every class should
participate meaningfully.
The diversity in communities and classrooms should be treated as an advantage that can help all learners in
Massachusetts’ schools. The presence of diverse learners in Massachusetts classrooms presents an opportunity
for all students and teachers to learn about the rest of the world and appreciate the talents and culture of each
individual. Since different cultures sometimes use alternative mathematical strategies or perceive the
relationships of objects and events in the world in ways other than the mainstream culture, their strategies and
understandings can enrich the understanding of all students. For example, Cambodian children learn a different
algorithm for division. If given the opportunity to explain their method to the rest of the class, then everyone
broadens their cultural experiences, deepens their understanding of the concept of division, and recognizes the
varied approaches to mathematics.
If the women, minorities, and individuals with disabilities who have made important contributions to the field
of mathematics and embarked on careers that utilize mathematics are presented as role models, then all students
see for themselves its practical applications. Knowing that increased opportunities await students who are
mathematically empowered, families, educators, and communities should encourage students to continue their
mathematics learning through grade twelve and beyond. Guidance counselors, students, and families should be
fully aware of the impact that mathematics has upon future access to higher education and employment
opportunities.
Mathematics For All
November 2, 2000
Page 14 of 55
Technology can help achieve equity in the classroom. With the allowance of graphing calculators on the
Scholastic Aptitude Tests for mathematics, physics, and chemistry and the requirement for the Advanced
Placement Test in Calculus, it is critical that all students who are preparing for these examinations gain
expertise in learning to use them. “It is important that all students have opportunities to use technology in
appropriate ways so that they have access to interesting and important ideas. Access to technology must not
become yet another dimension of educational inequality.” (NCTM 2000, p. 14)
Mathematics For All
November 2, 2000
Page 15 of 55
Guiding Principle II: Curriculum
Mathematics curriculum should be coherent and focused on important mathematics
and well articulated across the grades.
To help us understand the world, we draw upon a knowledge base that spans disciplines and experiences,
forming networks of thoughts and ideas. Students may relate their knowledge of functions to mechanics,
patterns to music, or statistics to the economy in Massachusetts. In all cases, they are connecting their
understanding of mathematics with other disciplines and their world.
A move toward connecting topics within the mathematics curriculum is recommended. Beginning with
preKindergarten, this approach to mathematics might include activities that combine sorting, measurement,
estimation, and geometry.
In middle schools and high schools, it will mean helping students make connections between algebra and
geometry, but also among ideas from discrete mathematics, statistics, and probability. For all, it will mean
establishing connections between mathematics and daily life at home, at work, and in the community.
A coherent mathematics curriculum gives students a more accurate picture of the nature of mathematics,
contextualizing the essential connections among various fields of mathematics. Connecting the domains of
mathematics encourages students to approach problem solving in more than one way, making them more
powerful problem solvers. Students will be able to solve problems numerically, algebraically, and graphically.
The use of technology and computer software facilitates connections. For example, graphing calculators make
it possible to switch from equations to graphs to data analysis. If a problem can be approached either visually
or numerically, then it may be more accessible to a visual learner struggling with abstraction. As solutions are
shared, these same visual learners will have the opportunity to explore the problem numerically.
“Learning mathematics involves accumulating ideas and building successively deeper and more refined
understanding. A School mathematics curriculum should provide a road map that helps teachers guide students
to increasing levels of sophistication and depths of knowledge. Such guidance requires a well-articulated
curriculum so that teachers at each level understand the mathematics that has been studied by students at the
previous level and what is to be the focus at successive levels.” (NCTM 2000, p. 16)
One of the most complex and difficult tasks for teachers, schools, and districts will be how to promote the
achievement of mathematical competence for all students. Some considerations on aligning curriculum to
individual student needs include:

High expectations and standards should be established for all students, including those with gaps in
their knowledge bases.

Students should be encouraged to achieve their highest potential in mathematics.

Students learn mathematics at different rates, and different students' interests in mathematics vary.
Support should be available for all students based on individual needs. Appropriate opportunities for
enrichment and advancement need to be provided for students at all achievement levels. The mathematics
should be challenging and expectations for all students should be high.
Mathematics For All
November 2, 2000
Page 16 of 55
Guiding Principle III: Learning
Students explore mathematical ideas in ways that maintain their enjoyment of and curiosity
about mathematics, help them develop depth of understanding,
and reflect real-world applications.
Though math learning follows a certain progression, it is not a purely linear process, but is
recursive, with children needing to rediscover and refine "old" concepts and skills as they
build "new" ones. Also, as with all learning, the development of math skills is unique to each
child. -- Early Childhood Today
Experimenting with ideas, inventing constructs, and exploring our curiosities are at the foundation of all
learning, including mathematics. A rich matrix of ideas should be explored thoroughly throughout each
academic year. Students should have regular opportunities to revisit important mathematical ideas throughout
each school year and from one year to the next.
If students are to develop mathematical understanding, then they should engage in tasks of inquiry, reasoning,
and problem solving that reflect real-world mathematical practice. In addition, hands-on exploration can
deepen understanding of abstract concepts by encouraging the practice of process skills and communication,
and allowing for reflective thinking.
Students learn best when they can connect their classroom learning to real-life experiences, and when they can
experience the same concept or idea in multiple contexts. The vision for mathematics in Massachusetts will
become reality only if students deepen their understanding of mathematics by means of activities,
investigations, and projects that promote inquiry, discovery, and mastery.
Investigations that a teacher introduces to a class should target important mathematical ideas, illuminate the
connections among mathematical ideas, and identify relationships between the ideas introduced in the
investigation and the concepts with which students are already familiar. It makes sense to embed problems or
investigations in and draw resources from the various cultures and backgrounds of students. The questions that
follow are suggested as one way to help teachers plan investigations.
Questions to consider when planning an investigation:

Have I identified and defined the mathematical content of the investigation, activity, or project?

How does the investigation address the learning styles and diverse backgrounds of all students within
the classroom?

Do I have a plan to initiate thoughtful discussion of or reflection on the concepts explored and of the
relationships uncovered in the process to help students clarify their understanding and integrate it fully
into their existing network of ideas?

Have I carefully compared the network of ideas included in the curriculum with the students'
knowledge?

Have I allowed time to note discrepancies, misunderstandings, and gaps in students' knowledge as well
as evidence of learning?

How is the investigation designed to test students' false assumption, confirm accurate findings, and
extend the students' knowledge?
Mathematics For All
November 2, 2000
Page 17 of 55
Guiding Principle IV: Teaching
Mathematics instruction involves understanding what students know and need to learn
and then engaging them to learn it well.
Developing mathematical competence is a complex process. The mathematics that students learn depends not
only on what is taught, but also on how it is taught. Curriculum cannot be separated from the instructional
practices used to teach it. Instructional strategies should encourage students to engage intellectually with
important mathematical ideas, to embrace the aesthetic value of mathematics, and to use mathematical
principles to solve problems in their daily lives.
Asking the right questions of students promotes creative thinking, prompting them to look deeper into their
imaginations. Students can be encouraged to reflect on their learning and articulate their reasoning through
questions such as:

How did you work through this problem?

Why did you choose this particular strategy to solve the problem?

Are there other ways? Can you think of them?

How can you be sure you have the correct solution?

Could there be more than one correct solution?

How can you convince me that your solution makes sense?
Teachers must have high expectations for all students and be familiar with each student's knowledge base in
order to plan developmentally appropriate work. The intellectual, social, and emotional development of
students should guide our choices of mathematics experiences for our students.
Working together in teams and groups enhances mathematical learning, helps students communicate
effectively, and develops social and mathematical skills. Students deepen their understanding of mathematics
as they interact with the ideas, theories, and opinions of their peers and teachers. Being able to communicate
mathematical ideas in a variety of ways helps students to "develop, test, and evaluate possible solutions" as
suggested in the Massachusetts Common Core of Learning. Teamwork encourages members to interact with
others, enhances self-assessment, encourages the exploration of multiple strategies, and helps prepare students
to be members of the workforce. The variation offered by group work leads to enriched solutions and offers an
ideal venue for informal assessment.
“Opportunities to reflect on and refine instructional practices are crucial in the vision of school mathematics
outlined here… Collaborating with colleagues regularly to observe, analyze, and discuss teaching and students’
thinking or to do a “lesson study” is a powerful, yet neglected, form of professional development in American
schools (Stigler and Hiebert 1999)3. The work and time of teachers must be structured to allow and support
professional development that will benefit them and their students.” (NCTM 2000, p. 19)
Stigler, James W., and James Hiebert. The Teaching Gap: Best Ideas from the World’s Teachers for Improving
Education in the Classroom. New York: The Free Press, 1999.
3
Mathematics For All
November 2, 2000
Page 18 of 55
Guiding Principle V: Assessment
Mathematics assessment is a multifaceted tool that monitors student performance, improves
instruction, enhances learning, and encourages student self-reflection.
The assessment should examine the extent to which students have integrated and made sense of
information, whether they can apply it to situations that require reasoning and creative
thinking, and whether they can use mathematics to communicate their ideas. -- National
Council of Teachers of Mathematics
The goal of classroom assessment is to provide information about students' evolving mathematical
understanding, skills, and knowledge so that teachers can give feedback to students and make decisions about
where to go next with their instruction. Mathematics assessment has been primarily short answers to short
questions. While standard assessment is one method of evaluation, a broader interpretation of mathematics
suggests that assessment must take on many new dimensions.
Performance-based assessments are especially congruent with the goals of mathematics instruction. The three
types of performance-based assessments discussed here are open-ended written assessments, portfolios, and
observation.
Open-ended written assessments present questions to students that invite multiple approaches to problems,
allow for creative expression of mathematical ideas, and encourage comparative analysis and reflection. By
soliciting written responses, students are encouraged to communicate their strategies, develop their hypotheses,
and explain their solutions by using prose, graphs, or drawings. An example of an open-ended question can be
as simple as asking students to explain their reasoning or to justify their answers, or as multifaceted as asking
them to design and conduct their own probability experiments.
Portfolio assessments imply that teachers have worked with students to establish individual criteria for selecting
work for placement in a portfolio and judging its merit. Charts, models, constructions, and students' reflections
on their work can all be included within mathematics portfolios. The contents of a mathematics portfolio
should be indicative of each student's abilities and understanding, representative of his efforts, and indicative of
progress over a period of time.
Observation as a means of assessment reflects a student's appreciation of mathematics, or the strategies he
commonly employs to solve problems, or perhaps his preferred learning style. Formal observations are preplanned, target specific mathematical skills, and refer to established criteria. These criteria for performance are
usually expressed within a range and made known to the student.
Mathematics For All
November 2, 2000
Page 19 of 55
Guiding Principle VI: Technology
Technology is an essential tool for effective mathematics education.
All students should use computers and other technologies to obtain, organize and communicate
information and to solve problems. -- Massachusetts Common Core of Learning
Students in all courses need access to tools for learning mathematics. These tools include measuring
instruments, manipulatives, graphing calculators, and computers. The calculation powers of computers and
calculators allow students to solve more complex problems but do not replace the thinking that underlies
mathematical operations. Computer software for modeling and visualization of mathematical ideas such as
statistics and probability or fractals and chaos can open a whole new world to students and help them connect
these mathematical ideas to their language and symbol systems.
"Students at one Massachusetts high school explore open-ended lab projects, using a geometry
construction software program that enables them to collect data, make observations, and develop
conjectures. They must support their findings by writing lab reports that summarize the exploratory
process and conclusions they have drawn. This makes the transition to deductive proof a much more
natural extension of the learning process. Students are encouraged to construct and justify their own
understanding of the geometry explored when they participate in a follow-up discussion, facilitated by
the teacher." (The Switched On Classroom, Massachusetts Software Council, 1994, p. 5-9)
Technology tools, when integrated in a mathematics program, raise the level of mathematics to which students
can be exposed, improve their self-confidence, and facilitate increased student-teacher interactions. The
availability of calculators, computers, and other technology has changed forever the way that people are able to
think about and do mathematics. New technologies have changed our culture into an "information society."
These changes have "transformed both the aspects of mathematics that need to be transmitted to students and
the concepts and procedures they must master if they are to be self-fulfilled, productive citizens." (Achieving
Mathematical Power, 1996) Some mathematics becomes more important because technology requires it, some
becomes less important because technology replaces it, and some becomes possible because technology allows
it. This integration of technology in our global society today impacts the lives of our students tomorrow.
When technology is implemented in our schools, change occurs. Electronic formats enable
access to information from almost anywhere in the world, crossing age groups and cultures
with ease…. Students move into the world possessing the skills to gather, manage, and
assimilate the vast resources of information at their fingertips. -- The Switched-on Classroom,
Massachusetts Software Council
For students with special needs, technology can be especially helpful in assisting students in regular and special
classrooms, the home, and the community. New software, hardware, and assistive devices can all be used to
help students succeed. Technology can enhance a student's access to the curriculum, increase opportunities to
interact with peers, and increase avenues of communication.
Technology facilitates students in communicating ideas within the classroom and in searching for information
in external databases. Indeed, the Internet has become an important supplement to library resources.
Technology can be especially helpful in assisting students with special needs in regular and special classrooms,
at home, and in the community. However, appropriate use of calculators is essential, and should not be used as
a replacement for basic understandings and skills. (NCTM 2000, p. 25 and p. 33)
Technology can be used to connect topics in mathematics, such as algebra, geometry and data analysis. It can
be used to help secondary students model and solve complex problems, instead of being limited to relatively
simple situations. (NCTM 2000, p. 26)
Mathematics For All
November 2, 2000
Page 20 of 55
Guiding Principle VII Lifelong Learning
Mathematics learning is a lifelong process that begins and continues in the
home and extends to school and community settings.
The need to make sense of the world begins before kindergarten and continues beyond formal schooling. PreKindergarten students begin to form ideas about mathematics as part of the natural process of exploring their
world. Building with blocks gives them an opportunity to begin developing an understanding of shape, size,
position, and symmetry. Gathering items such as rocks, shells, toy cars, or erasers for their collections leads to
discovery and exploration of patterns and classification. Such informal explorations are important
developmental precursors to an understanding of mathematics. They are the beginning of lifelong skills that
enable us to learn more abstractly.
Preschool activities give children opportunities to solve problems and discover information for themselves in an
environment where they can explore freely and safely. Young children and school-age students need repeated
experiences exploring materials, time to talk about their experiences, and freedom to experiment and learn from
each other. In addition to the mathematics that students learn in pre-Kindergarten through grade twelve
classrooms, there are other arenas in which the learning of mathematics can be strengthened.
Community programs, museums, and businesses can provide valuable learning experiences that enhance and
extend classroom activities. Careful planning in these contexts will provide opportunities for developing
students' curiosity, creating skepticism, and promoting self-esteem.
Adult mathematics learners often seek further education to meet a specific goal, perhaps to advance their
career, or to help their children, or for self improvement. Adult learners should be active participants in
defining personal learning objectives and deciding measures of success. Instruction should include
opportunities to question, discuss, and write about ideas..
Mathematics For All
November 2, 2000
Page 21 of 55
Strand Overviews
Strand 1: Number Sense and Operations
The study of numbers and operations is the cornerstone of the mathematics curriculum. Learning what
numbers mean, how they may be represented, relationships among them, and computations with them are
central to developing number sense. Students with number sense have the ability to decompose numbers
naturally, use particular numbers like 100 or 2 as referents, use the relationships among arithmetic operations to
solve problems, understand the base-ten number system, estimate, make sense of numbers, and recognize the
relative and absolute magnitude of numbers. (Sowder 1992)4
Research in developmental psychology and in mathematics education has shown that young children have a
great deal of informal knowledge of mathematics. As early as age three, children begin counting and
quantifying, and demonstrate an eagerness to do so. Capitalizing on this informal knowledge and interest,
education in the early years focuses on developing children’s facility with oral counting and recognition of
numerals and word names for numbers. Experience with counting naturally extends to quantification. Children
count objects and learn that the sizes, shapes, positions, or purposes of objects do not affect the total number of
objects in a group. One-to-one correspondence, with its matching of elements between two sets, provides the
foundation for the comparison of groups and gives meaning to counting. Combining and partitioning groups of
objects set the stage for operations with whole numbers, and the identification of equal parts of groups.
In the early elementary grades, students count and compute with whole numbers, learn different meanings of
the operations and relationships among them, and apply the operations to the solutions of problems. It is
important that students develop an understanding that numbers can be decomposed and used to solve problems
in many different ways. As they progress through the grades, students compute with multi-digit numbers,
estimate to judge the reasonableness of results of computations, and use concrete objects and diagrams to
model operations with fractions, mixed numbers, and decimals. By the end of their elementary school years,
students should be able to solve problems involving whole number computation, choose operations
appropriately, and understand the relationship of operations to one another. They should be able to choose
between appropriate methods (mental, paper and pencil, calculator) for solving problems, and estimate a
reasonable result for a problem. Students should develop a range of computational estimation strategies
including flexible rounding, the use of benchmarks, and front-end strategies.
All children need to develop computational fluency – having and using methods for computing accurately and
efficiently. “Fluency might be manifested in using a combination of mental strategies and jottings on paper or
using an algorithm with paper and pencil, particularly when numbers are large, to produce accurate results
quickly. It is important for students to have many opportunities to develop and explain strategies for solving
computational problems. Regardless of the particular method used, students should be able to explain their
method, understand that many methods exist, and see the usefulness of methods that are efficient, accurate, and
general…. Computational fluency should develop in tandem with understanding of the role and meaning of
arithmetic operations in number systems…” (NCTM 2000, p. 32)
“Researchers and experienced teachers alike have found that when children in the elementary grades are
encouraged to develop, record, explain, and critique one another’s strategies for solving computational
problems, a number of important kinds of learning can occur.... the efficiency of various strategies (as well as
their generalizability) can be discussed…. [E]xperience suggests that in classes focused on the development
Sowder, Judith T. “Making Sense of Numbers in School Mathematics.” An Analysis of Arithmetic for Mathematics
Teaching, edited by Gaea Leinhardt, Ralph Putman, and Rosemary A. Hattrup, pp. 1-51. Hillsdale, N.J.: Lawrence
Erlbaum Associates, 1992.
4
Mathematics For All
November 2, 2000
Page 22 of 55
and discussion of strategies, various ‘standard’ algorithms either arise naturally or can be introduced by the
teacher as appropriate.” (NCTM 2000, p.35) “.… [W]hen (preK - 2) students compute with strategies they
invent or choose because they are meaningful, their learning tends to be robust – they are able to remember and
apply their knowledge.” (NCTM 2000, p. 86) “Meaningful practice is necessary to develop fluency with basic
number combinations and strategies with multi-digit numbers…. Practice needs to be motivating and
systematic if students are to develop computational fluency, whether mentally, with manipulative materials, or
with paper and pencil. ... Practice should be purposeful and should focus on developing thinking strategies and
a knowledge of number relationships rather than drill isolated facts.” (NCTM 2000, p. 87)
“As students move from third to fifth grade, they should consolidate and practice a small number of
computational algorithms for addition, subtraction, multiplication, and division that they understand well and
can use routinely. ... Having access to more than one method for each operation allows students to choose an
approach that best fits the numbers in a particular problem. ... The conventional algorithms for multiplication
and division should be investigated in grades 3-5 as one efficient way to calculate. Regardless of the particular
algorithm used, students should be able to explain their method and should understand that many methods exist.
They should also recognize the need to develop efficient and accurate methods.” (NCTM 2000, p. 155). “As
students acquire conceptual grounding related to rational numbers, they should begin to solve problems using
strategies they develop or adapt from their whole-number work. At these (3-5) grades, the emphasis should not
be on developing general procedures to solve all decimal and fraction problems. Rather, students should
generate solutions that are based on number sense and properties of the operations and use a variety of models
or representations.” (NCTM 2000, p. 155)
In the middle grades, students should deepen their understanding and become proficient in solving problems
with fractions, decimals, percents and integers. At this level, using many different models can develop a sound
understanding of rational numbers and their different representations. Richly contextualized problems give
students the opportunity to gain facility with rational numbers and proportionality, as well as to connect their
learning to other topics. Understanding of proportionality develops as students use numbers, tables, graphs,
and equations to represent quantities and their relationships to one another.
Work with whole numbers continues in the middle grades as students study number theory and solve problems
and reason about factors, multiples, prime numbers, and divisibility.
At the high school level, understanding systems of numbers is enhanced through informal and formal
exploration of real numbers and computations with them. This work should form the basis for their work in
finding solutions for various types of equations. They should understand the difference between rational and
irrational numbers, and that the irrationals can only be approximated with fractions or repeating or terminating
decimals. Thereafter, students investigate complex numbers and relationships between the real and complex
numbers. Students expand their knowledge of counting techniques and apply those techniques to the solution of
problems.
As students develop competence with numbers and computation, they construct the scaffolding necessary to
build an understanding of number systems. Students not only compute and solve problems with different types
of numbers, but also explore the properties of operations on these numbers. Through investigation of
relationships among whole numbers, integers, rational numbers, real numbers, and complex numbers, students
gain an understanding of the structure of our number system.
Technology in the Number Sense and Operations strand is used to facilitate investigation of mathematical
concepts, skills, and strategies. Calculators and computers enhance students’ abilities to explore relationships
among different sets of numbers (e.g., the relationship between fractions and decimals, fractions and percents,
and decimals and percents); investigate alternative computational methods (e.g., generating the product of a
pair of multi-digit numbers on a calculator when the multiplication key cannot be used); verify results of
Mathematics For All
November 2, 2000
Page 23 of 55
computations done with other tools; compute with very large and very small numbers using numbers in
scientific notation form; and learn the rule for the Order of Operations.
Mathematics For All
November 2, 2000
Page 24 of 55
Strand 2: Patterns, Functions, and Algebra
"Mathematics is an exploratory science that seeks to understand every kind of pattern-patterns that occur in nature, patterns invented by the human mind, and even patterns created
by other patterns. To grow mathematically, children must be exposed to a rich variety of
patterns appropriate to their own lives through which they can see variety, regularity, and
interconnections." -- Lynn Arthur Steen, On the Shoulders of Giants
Patterns, Relations, and Functions are integral to the study of mathematics. Recognizing patterns and
describing their relations mathematically, by using geometry, number sequences, and functions, helps us
interact with and make sense of our world. To appreciate fully the intrinsic value of such pleasures as poetry,
art, music, plants, and animals, lifelong learners should know the mathematics of patterns and use mathematical
representations to describe them. The terminology of patterns, functions, and algebra has become a part of our
culture. Headlines and news reports speak of exponential growth of the national debt, a variable rate mortgage,
or a balanced budget.
"Algebraic competence is important in adult life, both on the job and as preparation for post secondary
education. All students should learn algebra." (NCTM 2000, p. 37)
Concepts for algebra are first formulated in preK. By the end of eighth grade, algebraic concepts should be
well developed. According to a US Department of Education, "Students who plan to take advanced
mathematics and science courses during high school and begin to study algebra during middle school are at a
clear advantage… Increasingly, schools are covering these rigorous content areas in courses that integrate
algebra, geometry and other areas of mathematics such as statistics and probability, rather than teaching each
separately." (Mathematics Equals Opportunity, U.S. Department of Education White Paper, October, 1997)
In order for students to become adept at using algebra as a problem solving tool, it is important for them to
"understand the concepts of algebra, the structures and principles that govern the manipulation of symbols, and
how the symbols themselves can be used for recording ideas and gaining insights into situations. Computer
(and calculator) technologies today can produce graphs of functions, perform operations on symbols, and
instantaneously do calculations on columns of data. Students need to learn how to interpret technological
representations and how to use the technology effectively and wisely." (NCTM 2000, p. 37)
Patterns and functions are the building blocks for transformational geometry, algebra, discrete mathematics,
trigonometry, and calculus. Underpinnings of calculus can be seen in algebra and geometry during earlier
grades. Older students will explore ideas about continuity, discontinuity, maximum and minimum. Real world
optimization problems, such as those related to maximum profit earned or minimum height achieved, provide
an opportunity to see the relevance of these concepts. Graphing calculators and computer software with
spreadsheet and graphics capabilities are ideal resources to use as students learn about functions. The meaning
of domain, range, roots, optimum values, periodicity, and other terms come alive when experienced through
technology. The use of technology such as graphing calculators enhances the development of students’ skills in
moving readily among symbolic, numeric, and graphic representations of functions and other relations.
Mathematics For All
November 2, 2000
Page 25 of 55
Strand 3: Geometry
The study of geometry is the study of shapes, both two and three-dimensional. It is also the study of
relationships, properties, attributes; a way to represent and analyze mathematical situations, and a route to
modeling real world situations so that they can be analyzed and measured.
Geometry offers students many opportunities for problem solving and reasoning, for making connections and
representations and for communicating with others about mathematics. Young students reason as they describe
their sorting and classifying of shapes and objects while secondary students use formal deductive reasoning to
prove congruence or similarity. Middle grade teachers may ask children to model a multiplication problem
with a rectangle, while middle school students model algebraic relationships in a similar fashion. Many of the
problem-solving strategies students develop as they move through the study of mathematics involve geometric
models, pictures, drawings and diagrams, as well as spatial visualization.
Young children bring a knowledge of shapes and space to school, and can begin their formal knowledge of
geometry by investigating and exploring shapes that can be held, stacked and put together to form new shapes.
They can be encouraged to observe various properties of shapes and sort them according to those properties.
They can learn to use the language of relative position, such as under, between, next to, after. They can
observe how objects and shapes change when transformed through the use of mirrors and folded paper cutting.
Teachers can use the natural curiosity and real-world experience of early childhood to help students build their
own mathematical foundation in geometry.
Later in elementary school, children use their informal experiential knowledge to more formally reason about
shapes. They can draw shapes, make arguments about their attributes and properties and relationships, and
begin to talk about the components of shapes – sides, angles, vertices, edges, faces. Understanding and
reasoning becomes more abstract as teachers encourage more precise descriptions using more precise language.
Students can begin to develop the idea of congruence through describing the motions that would transform one
shape into another. They also begin to use the coordinate plane to locate and to measure.
Middle school students use more formal reasoning to develop understanding of geometry. They explore and
make conjectures about similarity and congruence, use the coordinate plane more analytically to prove
congruence. Through net drawings and perspective drawings they explore the relationship between threedimensional objects and their two-dimensional representations. In high school, students use the skills and
concepts they’ve learned to analyze, reason, represent and model. They can formalize their reasoning skills to
prove the conjectures they’ve made using deductive reasoning. They can use coordinate geometry to analyze
mathematical situations and make arguments. They can model real life situations using geometry. Simulation,
drawing, and other software are useful tools for the exploration and development of geometric concepts in
middle and high school.
Geometry is often an area of mathematical strength and interest for students who have found the study of
mathematics difficult. Investigating problems using geometry can help all students grow in mathematical
understanding and sophistication, and form an important bridge between mathematics and other disciplines and
interests.
Mathematics For All
November 2, 2000
Page 26 of 55
Strand 4: Data Analysis, Statistics and Probability
All students should analyze, develop and act on informed opinions about current economic,
environmental, political and social issues affecting Massachusetts, the United States and the
world. -- Massachusetts Common Core of Learning
Statistics and probability confront us every day. The ability to understand variability and uncertainty is
necessary every time one reads the results of a Gallup poll or a report on the latest medical research findings.
If we are to interpret the arguments made by those on each side of an issue and make our own informed
decisions, then we must rely on our understanding of statistical inference, uses and misuses of statistics, and
probability.
Sorting and classifying by attributes are likely to be the first formal introduction to statistics that a student
encounters. At a very young age, students begin to describe, analyze, evaluate, and make decisions about the
attributes of familiar items such as blocks, shapes, or clothing. Students in early grades should explore
probability and data analysis by working with one variable and learning to make, read, and interpret simple
graphs. Students learn that a sample can be representative of a population. The more opportunities that students
have to do hands-on activities with probability and data, the better base they have from which they develop a
deeper understanding.
In their study of data and statistics, students shift their perspective from viewing data as a set of individual
pieces of information to an understanding of data as a coherent set with its own collective properties. This shift
is emphasized in the middle grades when students study characteristics of sets of data, including measures of
central tendency, and techniques for displaying these characteristics (e.g., stem-and-leaf plots). Students learn
how to select and construct representations most appropriate for the data and how to avoid misleading and
inappropriate representations. They also explore differences between theoretical and experimental probability.
They extend their data analysis and probability repertoire to include sampling bias and randomness.
Students in high school extend their knowledge to fitting nonlinear graphs to data. They study in depth
sampling methods and the role of sampling in making predictions and judging the validity of statistical claims.
The topics explored should be integrated with other mathematics courses.
The development of critical thinking in statistics should be emphasized. Students at all levels should formulate
appropriate questions; gather and explore data; organize and describe data, using graphs, charts, and tables;
interpret results; and develop a critical attitude toward the use of statistics. They should investigate real-life
problems that require them to employ sampling techniques. These investigations help them realize the relative
applicability of statistics to solving problems.
Technology has changed dramatically the way we deal with statistical data. Statisticians spend relatively more
time interpreting what their data suggest, using exploratory techniques, and relatively less time applying
standard inferential techniques. What computers have done for statisticians, inexpensive statistical calculators
and classroom software have done for students. More emphasis should be placed on interpretation of summary
statistics--both student-generated and from daily life.
Mathematics For All
November 2, 2000
Page 27 of 55
Strand 5: Measurement
Measurement, like no other aspect of science, is responsible for the historical development of mathematics. It
naturally lends itself to connections within mathematics and across other disciplines, including the social,
physical, and biological sciences. Measurement is best learned through direct applications or by being
embedded within other mathematical topics. A measurable attribute of an object is a characteristic that is most
readily quantified and compared. Many attributes, such as length, perimeter, area, volume, and angle measure,
come from the geometric realm. Other attributes are physical, such as temperature and mass. Still other
attributes are not readily measurable by direct means, as for example, speed and density.
Using their own informal units of measurement, students in preK through K make quantitative comparisons
between physical objects, e.g., which object is longer or shorter? which is lighter or heavier? which is warmer
or colder? Building on existing informal ideas, students in grades one and two become competent with
standard units of measurement. Students gain understanding of ratio and proportion in the middle grades, and
apply their newly found knowledge to making scale drawings and maps that accurately reflect the dimensions
of the landscape or the objects they represent. Greater familiarity with ratios enhances students’ understanding
of the derived attributes (speed, density, trigonometric ratios), their applications, and the use of conversion
factors to change a base unit in a measure.
At all levels, students develop respect for precision and accuracy by learning to select the tools and units of
measurement appropriate to the situation. They also learn to analyze possible and real errors in their
measurements and how those errors may be compounded in computations.
Mathematics For All
November 2, 2000
Page 28 of 55
GRADES preK-K
NUMBER SENSE AND OPERATIONS
NCTM LEARNING STANDARD: Understand numbers, ways of representing numbers,
relationships among numbers, and number systems
LEARNING EXPECTATIONS








Count forward by ones with understanding to at least 30.
Read numerals to 30.
Count backward from 10 with objects.
Compare two sets of objects using one-to-one correspondence.
Compare and order sets of up to 10 objects using appropriate terminology (same, more, less, etc.).
Develop understanding of relative position and use ordinal numbers to identify the position of objects in
sequences (first, second, up to fifth).
Connect numerals to the quantities they represent using various physical models and representations.
Develop understanding of whole and half by partitioning objects and groups.
NCTM LEARNING STANDARD: Understand meanings of operations and how they relate to one
another
LEARNING EXPECTATIONS



Develop understanding of the operations of addition and subtraction by modeling situations with objects and
drawings that involve joining together and taking apart quantities to 10.
Investigate the effects of addition and subtraction on whole numbers.
Explore the relationship between addition and subtraction.
NCTM LEARNING STANDARD: Compute fluently and make reasonable estimates
LEARNING EXPECTATIONS


Explore solving problem situations involving addition and subtraction of numbers up to 10.
Explore using estimation to judge reasonableness of results.
PATTERNS, FUNCTIONS AND ALGEBRA
NCTM LEARNING STANDARD: Understand patterns, relations, and functions
LEARNING EXPECTATIONS





Identify the attributes of objects as a foundation for sorting and classifying.
Sort and classify objects by color, shape, size, number, and other properties.
Recognize, reproduce, describe, extend, and create simple repeating patterns (color, rhythmic, shape, number, and
letter) and identify the unit being repeated.
Explore skip counting.
Explore the regular nature of patterns.
Mathematics For All
November 2, 2000
Page 29 of 55
GEOMETRY
NCTM LEARNING STANDARD: Analyze characteristics and properties of two- and threedimensional geometric shapes and develop mathematical arguments about geometric relationships
LEARNING EXPECTATIONS





Recognize, name, build, draw, compare, and sort two-dimensional shapes.
Recognize, name, build, compare, and sort three-dimensional shapes.
Describe attributes and parts of two- and three-dimensional shapes, e.g., sides, corners, edges, and faces.
Explore the relationship between two- and three- dimensional shapes, e.g., a pyramid has triangular faces.
Investigate symmetry of two- and three-dimensional shapes and constructions.
NCTM LEARNING STANDARD: Specify locations and describe spatial relationships using
coordinate geometry and other representational systems
LEARNING EXPECTATIONS

Identify positions of objects in space and use appropriate language (e.g., beside, inside, next to, close to, above,
below, apart) to describe and compare their relative positions.
NCTM LEARNING STANDARD: Use visualization, spatial reasoning, and geometric modeling to
solve problems
LEARNING EXPECTATIONS


Recognize geometric shapes and structures in the environment and specify their location.
Recognize shapes from different perspectives.
DATA ANALYSIS, STATISTICS, AND PROBABILITY
NCTM LEARNING STANDARD: Formulate questions that can be addressed with data and
collect, organize, and display relevant data to answer them
LEARNING EXPECTATIONS



Pose questions about themselves and their surroundings and gather data to answer the questions posed.
Sort and classify objects according to their attributes and organize data about the objects.
Represent data using concrete objects, pictures, numbers, lists, and simple graphs.
MEASUREMENT
NCTM LEARNING STANDARD: Understand measurable attributes of objects and the units,
systems, and processes of measurement
LEARNING EXPECTATIONS






Develop understanding of the attributes of length, volume, weight, area and time.
Use direct comparison to compare and order objects according to these attributes and use appropriate language,
e.g., longer, taller, shorter, same length; holds more, holds less, holds the same amount; heavier, lighter, same
weight.
Use nonstandard units to measure length, volume, weight, and area.
Explore time intervals (months/seasons, days of week, calendar, and clocks).
Identify U.S. coins.
Identify positions of events over time, e.g., earlier, later.
Mathematics For All
November 2, 2000
Page 30 of 55
NCTM LEARNING STANDARD: Apply appropriate techniques, tools, and formulas to determine
measurements
LEARNING EXPECTATIONS




Measure with multiple copies of units of the same size.
Explore making estimates of measurements.
Explore comparing and ordering two or more objects according to length, weight, area, and volume.
Explore and use standard units to measure and compare length.
Mathematics For All
November 2, 2000
Page 31 of 55
GRADES 1-2
NUMBER SENSE AND OPERATIONS
NCTM LEARNING STANDARD: Understand numbers, ways of representing numbers,
relationships among numbers, and number systems
LEARNING EXPECTATIONS











Count with understanding and recognize "how many" in sets of objects up to 100.
Count forward and backward from any given number.
Use multiple models to explore place value and the base ten number system.
Read and write (in numerals and words) whole numbers to 1000 and demonstrate an understanding of the value of
each digit.
Compare and order whole numbers using terms and symbols, e.g. less than, equal to, greater than (<, =, >).
Develop understanding of the relative position and magnitude of whole numbers.
Distinguish between ordinal (to tell which one) and cardinal (to tell how many) numbers.
Develop a sense of whole numbers, represent and use them in flexible ways including relating, composing, and
decomposing numbers.
Connect number words and numerals to the quantities they represent using various physical models and
representations.
Demonstrate an understanding of common fractions (1/2, 1/3, 1/4) as parts of wholes and as parts of groups.
Identify odd and even numbers and determine whether a set of objects has an odd or even number of elements.
NCTM LEARNING STANDARD: Understand meanings of operations and how they relate to one
another
LEARNING EXPECTATIONS




Demonstrate and apply various meanings of addition and subtraction (combining, finding a subset, separating,
comparing, equalizing, etc.).
Understand and use the inverse relationship between addition and subtraction to solve problems and check
solutions, e.g., 8 + 6 = 14 is related to 14 – 6 = 8.
Understand the effects of adding and subtracting whole numbers.
Explore concrete materials to investigate situations that relate to multiplication and division, e.g., equal groupings
of objects and sharing equally.
NCTM LEARNING STANDARD: Compute fluently and make reasonable estimates
LEARNING EXPECTATIONS




Develop, use and explain strategies that move toward efficiency for addition and subtraction of multi-digit whole
numbers.
Know addition combinations (addends to ten) and related subtraction combinations.
Use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and
pencil, and calculators.
Develop, use and explain strategies to estimate the reasonableness of answers involving addition and subtraction.
Mathematics For All
November 2, 2000
Page 32 of 55
PATTERNS, FUNCTIONS AND ALGEBRA
NCTM LEARNING STANDARD: Understand patterns, relations, and functions
LEARNING EXPECTATIONS





Sort, classify, and order objects by size, number, and other properties.
Recognize, reproduce, describe, extend, and create repeating patterns (color, rhythmic, shape, size, number, and
letter), identify the unit being repeated, and translate from one representation to another.
Use skip counting strategies to count by tens, fives, and twos up to at least 100.
Identify different number patterns on the hundred chart.
Analyze how both repeating and growing patterns are generated.
NCTM LEARNING STANDARD: Represent and analyze mathematical situations and structures
using algebraic symbols
LEARNING EXPECTATIONS




Develop understanding of and ability to use general principles and properties of operations such as
commutativity.
Use concrete, pictorial, written and verbal representations to develop an understanding of invented and
conventional symbolic notations.
Write number sentences to represent mathematical relationships using conventional symbols, e.g., +, -, =, <, >.
Construct and solve open sentences that have variables, e.g., 10 =  + 7, ∆ + ∆ = 10.
NCTM LEARNING STANDARD: Use mathematical models to represent and understand
quantitative relationships
LEARNING EXPECTATIONS



Model situations that involve the addition and subtraction of whole numbers, using objects, pictures, and
symbols.
Solve problems related to trading, including coin trades and measurement trades, e.g., pennies to nickels, quarts to
cups.
Investigate situations with variables as unknowns and as quantities that vary.
NCTM LEARNING STANDARD: Analyze change in various contexts
LEARNING EXPECTATIONS


Describe qualitative change such as a student growing taller.
Describe quantitative change such as a student growing two inches in one year.
GEOMETRY
NCTM LEARNING STANDARD: Analyze characteristics and properties of two- and threedimensional geometric shapes and develop mathematical arguments about geometric relationships
LEARNING EXPECTATIONS




Recognize, name, build, draw, compare, and sort two-and three-dimensional shapes including both polygonal (up
to 6 sides) and curved figures.
Describe attributes and parts of two- and three-dimensional shapes, e.g., sides, corners, edges, and faces.
Describe relationships between two- and three-dimensional shapes, e.g., a pyramid has triangular faces.
Investigate and predict the results of putting together and taking apart two- and three-dimensional shapes.
Mathematics For All
November 2, 2000
Page 33 of 55
NCTM LEARNING STANDARD: Specify locations and describe spatial relationships using
coordinate geometry and other representational systems
LEARNING EXPECTATIONS



Identify, describe, and interpret positions of objects in space and apply ideas about relative position.
Describe, name, and interpret direction and distance in navigating space and apply ideas about direction and
distance.
Find and name locations with simple relationships such as "near to" and in coordinate systems such as maps.
NCTM LEARNING STANDARD: Apply transformations and use symmetry to analyze
mathematical situations
LEARNING EXPECTATIONS



Recognize and apply slides, reflections and rotations.
Recognize and create shapes that have symmetry.
Investigate symmetry in two-dimensional shapes with mirrors or by paper folding.
NCTM LEARNING STANDARD: Use visualization, spatial reasoning, and geometric modeling to
solve problems
LEARNING EXPECTATIONS




Create mental images of geometric shapes using spatial memory and spatial visualization.
Recognize and represent shapes from different perspectives.
Relate geometric ideas to numbers, e.g., seeing rows in an array as a model of repeated addition.
Recognize geometric shapes and structures in the environment and specify their location.
DATA ANALYSIS, STATISTICS, AND PROBABILITY
NCTM LEARNING STANDARD: Formulate questions that can be addressed with data and
collect, organize, and display relevant data to answer them
LEARNING EXPECTATIONS



Pose questions about themselves and their surroundings and gather data by interviewing, surveying, and making
observations to answer the questions posed.
Sort and classify objects according to their attributes and organize data about the objects.
Represent data using concrete objects, pictures, numbers, tables, lists, tallies, graphs (bar graphs, pictographs),
and Venn diagrams.
NCTM LEARNING STANDARD: Select and use appropriate statistical methods to analyze data
LEARNING EXPECTATIONS

Describe and interpret data by drawing conclusions and making conjectures.
NCTM LEARNING STANDARD: Develop and evaluate inferences and predictions that are based
on data
LEARNING EXPECTATIONS

Discuss events related to students' experiences as likely or unlikely.
Mathematics For All
November 2, 2000
Page 34 of 55
NCTM LEARNING STANDARD: Understand and apply basic concepts of probability
LEARNING EXPECTATIONS


Investigate concepts of chance.
Make predictions about outcomes.
MEASUREMENT
NCTM LEARNING STANDARD: Understand measurable attributes of objects and the units,
systems, and processes of measurement
LEARNING EXPECTATIONS






Recognize the attributes of length, volume, weight, area, and time.
Use direct comparison to compare and order objects according to length, weight, area, and volume.
Measure and compare common objects using nonstandard and standard units of length.
Identify and use time intervals (days, months, weeks, hours, etc.)
Identify the value of U.S. coins and bills. Find and represent the value of a collection of coins and dollars up to
$5, using appropriate notation.
Select and use appropriate measurement tools.
NCTM LEARNING STANDARD: Apply appropriate techniques, tools, and formulas to determine
measurements
LEARNING EXPECTATIONS



Develop common referents to make comparisons and estimates of length, volume, weight, area, and time.
Measure with multiple copies of units of the same size.
Use repetition of a single unit to measure something larger than the unit.
Mathematics For All
November 2, 2000
Page 35 of 55
GRADES 3-4
NUMBER SENSE AND OPERATIONS
NCTM LEARNING STANDARD: Understand numbers, ways of representing numbers,
relationships among numbers, and number systems
LEARNING EXPECTATIONS









Exhibit an understanding of the place value structure of the base-ten number system by reading, modeling,
writing, and interpreting whole numbers up to 10,000; compare and order the numbers.
Recognize equivalent representations for the same number and generate them by decomposing and combining
numbers, e.g., 853 = 8 x 100 + 5 x 10 + 3; 853 = 85 x 10 + 3; 853 = 900 - 50 + 3.
Demonstrate an understanding of fractions as parts of unit wholes, as parts of groups, and as locations on number
lines.
Use visual models and benchmarks to recognize and generate equivalents of commonly used fractions and mixed
numbers (halves, thirds, fourths, fifths, sixths, eighths, tenths, and twelfths).
Use visual models, benchmarks (especially 1/2), and equivalent forms to compare and order commonly used
fractions.
Recognize and generate equivalent decimal forms of commonly used fractions less than one whole (halves,
quarters, fifths and tenths).
Explore numbers less than 0 by extending the number line and through familiar applications such as temperature.
Recognize classes of numbers, e.g., odds and evens, factors and multiples, and squares, to which a number may
belong, and identify numbers in those classes. Apply these concepts in the solution of problems.
Investigate prime and composite numbers and their relationship to factors and multiples.
NCTM LEARNING STANDARD: Understand meanings of operations and how they relate to one
another
LEARNING EXPECTATIONS






Use a variety of models to show understanding of multiplication and division of whole numbers, e.g., charts,
arrays, diagrams, physical models.
Investigate the effects of multiplying and dividing whole numbers.
Select and use appropriate operations (addition, subtraction, multiplication, division) to solve problems.
Identify and use relationships between operations, such as division as the inverse of multiplication, to solve
problems.
Apply concepts of commutative, associative, identity, and zero properties of operations on whole numbers in
problem situations, e.g., 37 X 46 = 46 x 37; (6 x 2) x 5 = 6 x (2 x 5); ∆ x 1 = ∆; ∆ x 0 = 0.
Investigate the concept of distributivity of multiplication over addition, e.g., 7 x 28 is equivalent to (7 x 20) + (7 x
8) or (7 x 30) - (7 x 2).
NCTM LEARNING STANDARD: Compute fluently and make reasonable estimates
LEARNING EXPECTATIONS




Know multiplication facts through 12 x 12 and related division facts. Use them to solve problems and mentally
compute related problems, e.g., 3 x 5 is related to 30 x 50, 300 x 5, and 30 x 500.
Add, subtract, and multiply (up to two digits by two digits) accurately and efficiently.
Demonstrate an understanding of and the ability to use a variety of procedures for addition and subtraction of
whole numbers, including conventional algorithms.
Divide (up to a three-digit whole number with a single-digit divisor) accurately and efficiently using a variety of
procedures. Interpret any remainders.
Mathematics For All
November 2, 2000
Page 36 of 55







Demonstrate an understanding of and the ability to use a variety of procedures for multiplication and division of
whole numbers.
Investigate conventional algorithms for multiplication and division of whole numbers.
Explore extending multiplication and division procedures to larger numbers. Evaluate the efficiency of the
procedures for working with larger numbers.
Explore relationships between various algorithms, including student generated, non-conventional, and
conventional, for solving the same problem.
Select and use a variety of strategies (e.g., front-end, rounding, and regrouping) to estimate the results of wholenumber computations and to judge the reasonableness of the answer.
Use visual models, benchmarks, concrete objects, and equivalent forms to add and subtract commonly used
fractions.
Select and use appropriate methods and tools for computing with whole numbers among mental computation,
estimation, calculators, and pencil-paper according to the nature of the computation
PATTERNS, FUNCTIONS AND ALGEBRA
NCTM LEARNING STANDARD: Understand patterns, relations, and functions
LEARNING EXPECTATIONS


Create, describe, and extend geometric and numeric patterns, including multiplication patterns, e.g., 30, 60, 90,
120; 3, 30, 300, 3000. Make predictions and form generalizations about the patterns.
Represent and analyze patterns and relationships, using words, models, tables, and graphs.
NCTM LEARNING STANDARD: Represent and analyze mathematical situations and structures
using algebraic symbols
LEARNING EXPECTATIONS





Represent the idea of a variable as an unknown quantity using a letter or a symbol, e.g., ∆, n.
Determine values of variables in simple equations, e.g., 4106 -  = 37;  - ∆ = 3; 4 + 5 =  + 3.
Use concrete materials and pictures to build an understanding of equality and inequality, and ways to maintain
these relations, e.g.,
How many stars will balance two squares?
Express mathematical relationships using equations, e.g., 4 + 5 = 7 + 2; 9 + 1 > 8 + 1; if ∆ = 5, then 6 + ∆ = 6 + 5
and 3 x ∆ = 3 x 5.
Explore the ways commutative, associative, distributive, identity, and zero properties are useful in computing
with whole numbers.
NCTM LEARNING STANDARD: Use mathematical models to represent and understand
quantitative relationships
LEARNING EXPECTATIONS


Model problem situations with objects and use representations such as pictures, graphs, tables, and equations to
draw conclusions.
Solve problems involving proportional relationships, including unit pricing, (e.g., four apples cost 80¢, so one
apple costs 20¢) and map interpretation (e.g., one inch represents five miles, so two inches represents ten miles).
Mathematics For All
November 2, 2000
Page 37 of 55
NCTM LEARNING STANDARD: Analyze change in various contexts
LEARNING EXPECTATIONS

Determine how change in one variable relates to a change in a second variable, e.g., input-output machines, data
tables.
GEOMETRY
NCTM LEARNING STANDARD: Analyze characteristics and properties of two- and threedimensional geometric shapes and develop mathematical arguments about geometric relationships
LEARNING EXPECTATIONS






Identify, compare, and analyze attributes of two- and three-dimensional shapes, e.g., sides, faces, angles, corners,
edges, diagonals, symmetry; and develop vocabulary to describe the attributes.
Recognize right angles and compare other angles to right angles, e.g., acute and obtuse angles.
Identify, describe, and draw intersecting, parallel, and perpendicular lines.
Classify two- and three-dimensional shapes according to their properties and develop definitions for classes of
shapes such as triangles, quadrilaterals, and pyramids.
Investigate, describe, and predict the results of subdividing, combining, and transforming shapes.
Investigate congruence and similarity.
NCTM LEARNING STANDARD: Specify locations and describe spatial relationships using
coordinate geometry and other representational systems
LEARNING EXPECTATIONS



Describe location and movement using common language and geometric vocabulary, e.g., directions from the
classroom to the gym.
Using ordered pairs, graph, locate, identify points, and describe paths in the first quadrant of the coordinate plane.
Use a two-dimensional grid system, such as a map, to locate positions representing actual places.
NCTM LEARNING STANDARD: Apply transformations and use symmetry to analyze
mathematical situations
LEARNING EXPECTATIONS



Predict and describe the results of sliding, flipping, and turning two-dimensional shapes.
Describe a motion or a series of motions that will show that two shapes are congruent.
Identify and describe line-symmetry in two-dimensional shapes and designs.
NCTM LEARNING STANDARD: Use visualization, spatial reasoning, and geometric modeling to
solve problems
LEARNING EXPECTATIONS






Build and draw geometric objects.
Create and describe mental images of objects, patterns, and paths.
Identify and build three-dimensional objects from two-dimensional representations of that object.
Investigate two-dimensional representations of three-dimensional objects.
Use geometric models to solve problems in other areas of mathematics, such as using arrays as models of
multiplication or area.
Explore geometric ideas and relationships as they apply to other disciplines and to problems that arise in the
classroom or in everyday life.
Mathematics For All
November 2, 2000
Page 38 of 55
DATA ANALYSIS, STATISTICS, AND PROBABILITY
NCTM LEARNING STANDARD: Formulate questions that can be addressed with data and
collect, organize, and display relevant data to answer them
LEARNING EXPECTATIONS





Design investigations to address a question and consider how data-collection methods affect the nature of the data
set.
Collect and organize data using observations, measurements, surveys, or experiments.
Represent data using tables and graphs such as line plots, bar graphs and line graphs.
Choose and construct representations that are appropriate for the data set.
Recognize the differences in representing categorical and numerical data.
NCTM LEARNING STANDARD: Select and use appropriate statistical methods to analyze data
LEARNING EXPECTATIONS



Describe the shape and important features of a set of numerical data, with an emphasis on how the data are
distributed, including the range, where the data are concentrated or sparse, and whether there are outliers.
Explore the concepts of median, mode, maximum, minimum, and range, and consider what each does and does
not indicate about the data set.
Compare related data sets, with emphasis on the range, center, and how the data are distributed.
NCTM LEARNING STANDARD: Develop and evaluate inferences and predictions that are based
on data
LEARNING EXPECTATIONS

Propose and justify conclusions and predictions based on the data.
NCTM LEARNING STANDARD: Understand and apply basic concepts of probability
LEARNING EXPECTATIONS




Describe events as likely or unlikely and discuss the degree of likelihood using such words as certain, likely,
unlikely, equally likely, and impossible.
Predict the likelihood of outcomes of simple experiments and test the predictions using concrete objects such as
counters, number cubes, spinners, or coins.
Record the probability of a specific outcome for a simple probability situation, e.g., ; probability is
3 out of 7 of drawing a black ball; 3/7.
List the number of possible combinations of objects from 3 sets, e.g., number of outfits from 3 shirts, 2 skirts, and
2 hats.
MEASUREMENT
NCTM LEARNING STANDARD: Understand measurable attributes of objects and the units,
systems, and processes of measurement
LEARNING EXPECTATIONS



Identify attributes such as length, area, weight, and volume, and select the appropriate type of unit for measuring
each attribute.
Use concrete objects to explore volume and surface area of rectangular prisms.
Demonstrate an understanding of the need for measuring with standard units and become familiar with standard
units in the customary and metric systems.
Mathematics For All
November 2, 2000
Page 39 of 55



Explore the inverse relationship between the size of the unit and the number of units.
Understand that measurements are approximations and investigate how differences in units affect precision.
Consider the degree of accuracy needed for different situations.
Carry out simple unit conversions within a system of measurement, e.g., hours to minutes, cents to dollars, yards
to feet or inches.
NCTM LEARNING STANDARD: Apply appropriate techniques, tools, and formulas to determine
measurements
LEARNING EXPECTATIONS








Develop strategies for estimating perimeters and areas of rectangles, triangles, and irregular shapes.
Develop strategies for finding the area of rectangles and right triangles.
Select and use appropriate standard units and tools to estimate, measure, and solve problems involving length,
area, volume, weight, time, and temperature.
Recognize a 90° angle and use it as a benchmark to estimate the size of other angles.
Identify common measurements of turns, e.g., 360° in one full turn, 180° in a half turn, and 90° in a quarter turn.
Investigate the use of protractors and angle rulers to measure angles.
Compute elapsed time, and make and interpret schedules.
Select and use benchmarks to estimate measurements.
Mathematics For All
November 2, 2000
Page 40 of 55
GRADES 5-6
NUMBER SENSE AND OPERATIONS
NCTM LEARNING STANDARD: Understand numbers, ways of representing numbers,
relationships among numbers, and number systems
LEARNING EXPECTATIONS











Use models, benchmarks, and equivalent forms to judge the size of fractions.
Explore numbers less than 0 through familiar applications.
Describe classes of numbers according to characteristics such as the nature of their factors.
Explore the use of ratios and proportions to represent quantitative relationships
Use factors, multiples, prime factorization, and relatively prime numbers to solve problems.
Demonstrate an understanding of place value to billions and thousandths.
Represent and compare very large (billions) and very small (thousandths) positive numbers in various forms.
Demonstrate an understanding of positive integer exponents, especially when used in powers of ten, e.g., 42, 105.
Demonstrate an understanding of fractions as a ratio of whole numbers, as parts of unit wholes, as parts of a
collection, and as locations on number lines.
Compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number
line.
Develop an understanding of and appropriately apply number theory concepts – including prime and composite
numbers, square numbers, prime factorization, greatest common factors, least common multiples, and divisibility
rules for 2, 3, 4, 5, 6, 9, 10 - to the solution of problems.
NCTM LEARNING STANDARD: Understand meanings of operations and how they relate to one
another
LEARNING EXPECTATIONS





Demonstrate an understanding of various meanings and effects of multiplication and division.
Demonstrate an understanding of the meaning and effects of arithmetic operations with fractions and decimals.
Use the associative and commutative properties of addition and multiplication and the distributive property of
multiplication over addition to simplify computations with whole numbers, fractions, and decimals.
Understand and use the inverse relationships of addition and subtraction, multiplication and division, to simplify
computations and solve problems.
Apply the Order of Operations for expressions involving addition, subtraction, multiplication, and division with
grouping symbols.
NCTM LEARNING STANDARD: Compute fluently and make reasonable estimates
LEARNING EXPECTATIONS






Develop fluency in adding, subtracting, multiplying and dividing whole numbers using paper and pencil and
mental computation.
Use visual models, benchmarks and equivalent forms to develop understanding of addition and subtraction of
commonly used fractions and decimals.
Select and use appropriate methods and tools for computing with whole numbers and decimals from among
mental computation, estimation, calculators, and paper and pencil according to the context and nature of the
computation and use the selected method or tool.
Develop and analyze algorithms for computing with fractions and decimals and develop fluency in their use.
Develop and use strategies to estimate the results of whole number and rational number computations and judge
the reasonableness of the results.
Develop fluency with addition, subtraction, multiplication and division of positive fractions and mixed numbers.
Mathematics For All
November 2, 2000
Page 41 of 55
PATTERNS, FUNCTIONS AND ALGEBRA
NCTM LEARNING STANDARD: Understand patterns, relations, and functions
LEARNING EXPECTATIONS

Represent, analyze and generate a variety of geometric and arithmetic patterns and progressions with tables,
graphs, words, and, when possible, symbolic rules. e.g., ABBCCC…; 1, 5, 9, 13…; 3, 9, 27…
NCTM LEARNING STANDARD: Represent and analyze mathematical situations and structures
using algebraic symbols
LEARNING EXPECTATIONS




Demonstrate an understanding of such properties as commutativity, associativity, and distributivity and use them
to compute.
Express mathematical relationships using equations.
Compute the value of variables in input/output tables,
Replace variables with given values and evaluate or simplify.
NCTM LEARNING STANDARD: Use mathematical models to represent and understand
quantitative relationships
LEARNING EXPECTATIONS


Model and solve contextualized problems and mathematical relationships with concrete models, tables, graphs,
and rules in words, and with symbols, e.g., input-output tables.
Explore situations with proportional relationships through models.
NCTM LEARNING STANDARD: Analyze change in various contexts
LEARNING EXPECTATIONS




Investigate how a change in one variable relates to a change in a second variable.
Identify and describe situations with constant or varying rates of change and compare them.
Use graphs to analyze the nature of changes in quantities in linear relationships.
Explore the concept of change, e.g., how a change in one variable affects a second variable, using physical
models.
GEOMETRY
NCTM LEARNING STANDARD: Analyze characteristics and properties of two- and threedimensional geometric shapes and develop mathematical arguments about geometric relationships
LEARNING EXPECTATIONS



Identify and compare polygons based on their attributes, including types of interior angles, perpendicular or
parallel sides, and congruence of sides and angles.
Classify two- and three-dimensional shapes based on their properties such as sides, edges, and faces.
Make and test conjectures about geometric properties and relationships, and develop logical arguments to justify
conclusions.
Mathematics For All
November 2, 2000
Page 42 of 55
NCTM LEARNING STANDARD: Specify locations and describe spatial relationships using
coordinate geometry and other representational systems
LEARNING EXPECTATIONS



Make and use coordinate systems to specify locations and identify the coordinates of points within the first
quadrant.
Explore graphing in all four quadrants of the Cartesian coordinate plane.
Find the distance between two points along horizontal lines or along vertical lines of a coordinate system.
NCTM LEARNING STANDARD: Apply transformations and use symmetry to analyze
mathematical situations
LEARNING EXPECTATIONS



Predict, describe and perform transformations on two-dimensional shapes.
Make and test conjectures on polygons that will tessellate.
Explore various types of symmetry.
NCTM LEARNING STANDARD: Use visualization, spatial reasoning, and geometric modeling to
solve problems
LEARNING EXPECTATIONS
 Draw two-dimensional objects with specific properties such as side lengths or angle measures.
 Investigate drawing shapes from different views and perspectives.
 Identify and build three-dimensional objects from their two-dimensional representations, including top view, side



view, and front view.
Experiment drawing a two-dimensional representation of a three-dimensional object.
Recognize geometric ideas and relationships and apply them to other disciplines such as art and science and to
problems that arise in the mathematics classroom or in everyday life.
Investigate networks to represent and solve problems.
DATA ANALYSIS, STATISTICS, AND PROBABILITY
NCTM LEARNING STANDARD: Formulate questions that can be addressed with data and
collect, organize, and display relevant data to answer them
LEARNING EXPECTATIONS



Formulate questions, design studies and collect data about a characteristic of two populations or different
characteristics among one population.
Select, create, and use appropriate graphical representations of data, including histograms, bar graphs, box plots,
line plots and scatterplots.
Investigate the use of circle graphs.
NCTM LEARNING STANDARD: Select and use appropriate statistical methods to analyze data
LEARNING EXPECTATIONS



Describe and compare data sets using the concepts of median, mean, mode, maximum, minimum, and range.
Describe and understand the correspondence between data sets and their representations, especially histograms,
bar graphs, line plots, box plots and scatterplots. Understand what various representations reveal about the data.
Compare representations of the same data and evaluate how well each representation shows important aspects of
the data.
Mathematics For All
November 2, 2000
Page 43 of 55
NCTM LEARNING STANDARD: Develop and evaluate inferences and predictions that are based
on data.
LEARNING EXPECTATIONS

Propose and justify conclusions and predictions that are based on data and design studies to further investigate the
conclusions or predictions.
NCTM LEARNING STANDARD: Understand and apply basic concepts of probability
LEARNING EXPECTATIONS




Describe the degrees of likelihood of events using terms such as likely, unlikely, certain and impossible.
Predict the probability of outcomes of simple experiments and test the predictions.
Use appropriate ratios between 0 and 1 to represent the probability of an event and associate the probability with
the likelihood of the event.
Compute probabilities for simple compound events, using a variety of methods including organized lists, tree
diagrams, and area models.
MEASUREMENT
NCTM LEARNING STANDARD: Understand measurable attributes of objects and the units,
systems, and processes of measurement
LEARNING EXPECTATIONS





Demonstrate an understanding of such attributes as length, area, weight, volume, and size of angle, and select an
appropriate unit for measuring each attribute.
Demonstrate an understanding of the need for measuring with standard units and become familiar with standard
units in the customary and metric systems.
Understand relationships among units within the same system (metric or customary) and carry out simple unit
conversions, such as centimeters to meters.
Understand that measurements are approximations and understand how differences in unit affect precision.
Explore what happens to measurements of a two-dimensional shape such as its perimeter and area when the shape
is changed in some way.
NCTM LEARNING STANDARD: Apply appropriate techniques, tools, and formulas to determine
measurements
LEARNING EXPECTATIONS









Select and use benchmarks to estimate measurements, e.g., the height of a doorknob is about 2 meters.
Find areas of triangles and parallelograms using models.
Develop strategies for estimating the perimeters and areas of irregular shapes.
Solve problems involving proportional relationships and units of measurement, e.g., manageable scale models.
Explore the relationships of the radius, diameter, and circumference of circles.
Develop an understanding of and use formulas to find areas of rectangles and related triangles and
parallelograms.
Develop strategies to determine the surface areas and volumes of rectangular solids.
Explore various models for finding the area of parallelograms, trapezoids and circles.
Explore volume and surface area of three-dimensional shapes such as prisms, pyramids, and cylinders.
Mathematics For All
November 2, 2000
Page 44 of 55
GRADES 7-8
NUMBER SENSE AND OPERATIONS
NCTM LEARNING STANDARD: Understand numbers, ways of representing numbers,
relationships among numbers, and number systems
LEARNING EXPECTATIONS









Demonstrate an understanding of multiple representations for quantitative relationships such as fractions to
decimals to percents, e.g. 3/2 = 1.5 = 150%
Demonstrate facility in using fractions, decimals and percents as appropriate in the solution of problems.
Compare and order fractions, decimals, and integers and locate correctly on number lines.
Demonstrate an understanding of and use ratios and proportions to represent quantitative relationships.
Use ratios and proportions in the solution of problems to include unit rates, scale factors, and rate of change.
Represent large numbers using exponential, scientific, and calculator notation appropriately.
Apply number theory concepts, including prime factorization and relatively prime numbers, to the solution of
problems.
Investigate and use negative integral exponents and their use in scientific and calculator notation.
Investigate absolute value as meaning distance on a number line.
NCTM LEARNING STANDARD: Understand meanings of operations and how they relate to one
another
LEARNING EXPECTATIONS





Demonstrate an understanding of the meaning and effects of arithmetic operations with fractions, decimals and
integers.
Demonstrate an understanding of and apply the Order of Operations to include non-negative exponents and
grouping symbols.
Compare, order, and apply frequently used irrational numbers, e.g., square root of 2, π.
Use the associative and commutative properties of addition and multiplication and the distributive property of
multiplication over addition to simplify computations with integers, fractions and decimals.
Demonstrate an understanding of and use the inverse relationships of addition and subtraction, multiplication and
division, and squaring and finding square roots to simplify computations and solve problems., e.g. multiplying by
½ or 0.5 is the same as dividing by 2.
NCTM LEARNING STANDARD: Compute fluently and make reasonable estimates
LEARNING EXPECTATIONS





Select appropriate methods and tools for computing with fractions and decimals from among mental computation,
estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected
methods.
Determine when an estimate rather than an exact answer is appropriate and apply in problem situations.
Develop and analyze algorithms for computing with fractions, decimals and integers and develop fluency in their
use.
Develop and adapt procedures for mental calculation and computational estimation with fractions, decimals,
percents and integers and judge the reasonableness of results.
Develop, analyze, and explain methods for solving problems involving proportions.
Mathematics For All
November 2, 2000
Page 45 of 55
PATTERNS, FUNCTIONS AND ALGEBRA
NCTM LEARNING STANDARD: Understand patterns, relations, and functions
LEARNING EXPECTATIONS



Extend, represent, analyze, and generalize a variety of patterns (numeric and visual) with tables, graphs, words,
and when possible, symbolic expressions.
Demonstrate an understanding of graphical, tabular, or symbolic representations for a given pattern and/or
relationship.
Identify functions as linear or nonlinear and contrast their properties from tables, graphs or equations.
NCTM LEARNING STANDARD: Represent and analyze mathematical situations and structures
using algebraic symbols
LEARNING EXPECTATIONS






Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the
meaning of slope and intercept.
Identify the slope of a line with its steepness and as a constant rate of change from its table of values, equation,
and graph.
Demonstrate an understanding of the roles of variables and constants within an equation representing a linear
function, e.g. y = mx + b.
Represent and solve linear equations and/or inequalities with one or two variables using models, symbols, and/or
graphs.
Use symbolic algebra to represent situations and solve problems, especially those involving linear relationships.
Explain and analyze – using words, pictures, graphs, charts, or equations – how a change in one variable results in
a change in another variable in functional relationships, e.g., A = πr 2 means that the area of a circle must increase
if the radius increases.
NCTM LEARNING STANDARD: Use mathematical models to represent and understand
quantitative relationships
LEARNING EXPECTATIONS


Use graphs, tables and linear equations or inequalities to model and analyze contextualized problems. Use
technology as appropriate.
Investigate the use of tables, graphs and equations to represent systems of linear functions to solve contextualized
problems. Use technology as appropriate.
NCTM LEARNING STANDARD: Analyze change in various contexts
LEARNING EXPECTATIONS
 Use tables and graphs to represent and compare linear growth patterns. In particular, compare rates of change and

intercepts.
Explore the similarities among linear, exponential, and quadratic growth patterns in analyzing problem situations.
Mathematics For All
November 2, 2000
Page 46 of 55
GEOMETRY
NCTM LEARNING STANDARD: Analyze characteristics and properties of two- and threedimensional geometric shapes and develop mathematical arguments about geometric relationships
LEARNING EXPECTATIONS






Demonstrate an understanding of the attributes of polygons and analyze their defining properties.
Classify polygons in terms of congruence and similarity, and apply these relationships to the solution of
problems.
Demonstrate an understanding of the relationships of angles formed by intersecting lines, to include parallel lines
cut by a transversal.
Explore visual and other proofs of the Pythagorean Theorem.
Investigate trigonometric ratios in right triangles.
Investigate right triangle relationships, such as those in 45-45-90 and 30-60-90 triangles.
NCTM LEARNING STANDARD: Specify locations and describe spatial relationships using
coordinate geometry and other representational systems
LEARNING EXPECTATIONS




Graph points in all four quadrants of the Cartesian coordinate plane and identify coordinates of points.
Investigate absolute value as representing distance and apply it to the solutions of problems.
Graph triangles and quadrilaterals on a coordinate plane and examine their properties using coordinates.
Explore properties of regular polygons as well as other triangles and quadrilaterals, associating slopes of lines
with parallel and perpendicular lines, and determining lengths of sides.
NCTM LEARNING STANDARD: Apply transformations and use symmetry to analyze
mathematical situations
LEARNING EXPECTATIONS




Explore the results of flips, turns, and slides, and investigate relationships among composition of transformations
using physical objects, tracing paper, mirrors, graph paper, and/or dynamic geometry software.
Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides,
and scaling.
Demonstrate an understanding of congruence as well as line and rotational symmetry using transformations.
Formulate and test conjectures about shapes that tessellate.
NCTM LEARNING STANDARD: Use visualization, spatial reasoning, and geometric modeling to
solve problems
LEARNING EXPECTATIONS



Demonstrate an understanding of the relationships among two- and three-dimensional objects to include twodimensional nets for three- dimensional objects.
Recognize and draw two-dimensional representations of three-dimensional objects.
Demonstrate an understanding of and use geometric models to explain numerical and algebraic relationships, e.g.,
triangular numbers represented by "steps” and algebraic identities and/or properties represented by area models.
Mathematics For All
November 2, 2000
Page 47 of 55
DATA ANALYSIS, STATISTICS, AND PROBABILITY
NCTM LEARNING STANDARD: Formulate questions that can be addressed with data and
collect, organize, and display relevant data to answer them
LEARNING EXPECTATIONS




Formulate questions about characteristics of populations, identify different ways of selecting a data sample, and
describe the characteristics and limitations of a data sample.
Identify different ways of selecting a sample, e.g. convenience sampling, responses to a survey, random sampling.
Select, create, interpret, and utilize various tabular and graphical representations of data, e.g., circle graphs, Venn
diagrams, scatterplots, stem-and-leaf, box plots, histograms, tables, and charts.
Explore the differences between continuous and discrete data and ways to represent them.
NCTM LEARNING STANDARD: Select and use appropriate statistical methods to analyze data
LEARNING EXPECTATIONS


Find, describe and interpret appropriate measures of central tendency (mean, median, and mode) and spread
(range) that represent a set of data.
Determine the appropriateness of different representations, graphical or tabular, for given data sets.
NCTM LEARNING STANDARD: Develop and evaluate inferences and predictions that are based
on data
LEARNING EXPECTATIONS



Determine and evaluate inferences based on the analysis of a sample set of data.
Make and defend conjectures about possible relationships of a sample, given a scatterplot of the sample data and
an approximate line of fit.
Use conjectures to formulate new questions and determine new studies to answer them.
NCTM LEARNING STANDARD: Understand and apply basic concepts of probability
LEARNING EXPECTATIONS




Understand and use appropriate terminology of probability, including describing complementary and mutually
exclusive events.
Use tree diagrams, tables, organized lists, and area models to compute probabilities for simple compound events,
e.g., multiple coin tosses or rolls of dice.
Investigate the relationship of experimental and theoretical probability.
Explore mathematical fairness in games.
MEASUREMENT
NCTM LEARNING STANDARD: Understand measurable attributes of objects and the units,
systems, and processes of measurement
LEARNING EXPECTATIONS


Select, convert within the same system of measurement, and use appropriate units of measurement or scale.
Develop an understanding of the relationship of customary and metric units and select units of appropriate size
and type to measure angles, perimeter, area, surface area, and volume.
Mathematics For All
November 2, 2000
Page 48 of 55
NCTM LEARNING STANDARD: Apply appropriate techniques, tools, and formulas to determine
measurements
LEARNING EXPECTATIONS





Develop and use formulas to determine the circumference of circles, and the area of triangles, parallelograms,
trapezoids, and circles and develop strategies to find the area of more complex shapes.
Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate
levels of precision.
Develop strategies to determine the surface area and volume of prisms, pyramids, and cylinders.
Use ratio and proportion, including scale factors, in the solution of problems.
Use models, graphs, and formulas to solve simple problems involving rates and derived measurements for such
attributes as velocity and density.
Mathematics For All
November 2, 2000
Page 49 of 55
GRADES 9-10
NUMBER SENSE AND OPERATIONS
NCTM LEARNING STANDARD: Understand numbers, ways of representing numbers,
relationships among numbers, and number systems
LEARNING EXPECTATIONS


Compare and contrast properties of sets of numbers within the real number system.
Demonstrate an understanding of absolute value on a number line.
NCTM LEARNING STANDARD: Compute fluently and make reasonable estimates
LEARNING EXPECTATIONS






Identify and use the properties of operations on real numbers, including the associative, commutative, and
distributive properties. Use the identity and inverse elements for the four basic operations.
Explain the density property of the set of rational numbers.
Apply operations with powers to evaluate or rewrite numerical expressions.
Find the approximate value of solutions to problems involving square roots.
Use estimation to judge the reasonableness of results of computations and of solutions to problems involving real
numbers.
Analyze relationships among various subsets of the real numbers (the whole numbers, the integers, the even
integers, the rational numbers, and the irrational numbers).
PATTERNS, FUNCTIONS AND ALGEBRA
NCTM LEARNING STANDARD: Understand patterns, relations, and functions
LEARNING EXPECTATIONS




Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including
iterative/recursive (e.g., Fibonnacci Numbers), linear, quadratic and exponential functions.
Explain the difference between linear and exponential growth.
Use tables, graphs, and words to analyze linear, quadratic and exponential functions, including zeros/roots,
domain and range.
Interpret the meaning of function values including f(x) notation, x- and y-intercepts, and domain and range in real
world problems.
NCTM LEARNING STANDARD: Represent and analyze mathematical situations and structures
using algebraic symbols
LEARNING EXPECTATIONS





Apply the Order of Operations and properties of addition and multiplication to rewrite algebraic expressions that
include exponents. Simplify expressions by rearranging and collecting terms.
Use algebraic, tabular and graphical methods, including technology when appropriate, to model and solve
systems of linear equations and inequalities.
Identify problem situations that can be modeled by linear equations and solve by applying appropriate graphical,
tabular, or symbolic methods. Describe relationships among the methods.
Investigate problem situations that lead to linear and quadratic equations (with real number solutions) and solve
by applying appropriate graphical, tabular or symbolic methods.
Recognize real-world situations involving both negative and positive constant rates of changes.
Mathematics For All
November 2, 2000
Page 50 of 55




Explore matrices and the operations of addition and multiplication of matrices.
Explore the use of matrices to represent situations with variable quantities and to solve systems of linear
equations.
Investigate recursive function notation.
Explore factoring more difficult trinomials, such as those where the quadratic coefficient does not equal one.
GEOMETRY
NCTM LEARNING STANDARD: Analyze characteristics and properties of two- and threedimensional geometric shapes and develop mathematical arguments about geometric relationships
LEARNING EXPECTATIONS






Identify figures using properties of sides, angles, and diagonals. Identify what type(s) of symmetry a figure has.
Recognize and solve problems involving angles formed by parallel lines cut by a transversal.
Identify major and minor arcs and recognize the relationship between central angle and arc measure.
Solve simple triangle problems using the triangle "angle sum" property.
Use spatial relationships such as parallel faces to identify and classify common solids, e.g., pyramids, prisms,
cones and cylinders.
Given measures of some parts of geometric figures; apply congruence and similarity properties to determine
measures of other parts, providing logical justification.
NCTM LEARNING STANDARD: Specify locations and describe spatial relationships using
coordinate geometry and other representational systems
LEARNING EXPECTATIONS


Draw geometric figures on a coordinate plane.
Calculate the midpoint and slope of a segment. Calculate the distance between two points. Apply the results to
the solutions of problems
NCTM LEARNING STANDARD: Apply transformations and use symmetry to analyze
mathematical situations
LEARNING EXPECTATIONS


Perform transformations (translations, reflections, rotations, scale changes, size changes, and combinations of
them) on figures in the coordinate plane.
Explore the effect of such transformations on the attributes of the original figure.
NCTM LEARNING STANDARD: Use visualization, spatial reasoning, and geometric modeling to
solve problems
LEARNING EXPECTATIONS





Given a figure, perform various constructions (parallel or perpendicular segments, congruent or similar figures)
using a compass, straightedge, and, where appropriate, other tools such as protractor or computer software. Make
conjectures about methods of construction. Justify the conjectures by logical arguments.
Demonstrate the ability to visualize solid objects and recognize their projections and cross sections.
Investigate the use of vertex-edge graphs to model and solve problems.
Recognize the relationships between the slopes of perpendicular and parallel lines.
Explore the properties of chords, tangents, and secants in solving problems.
Mathematics For All
November 2, 2000
Page 51 of 55
DATA ANALYSIS, STATISTICS, AND PROBABILITY
NCTM LEARNING STANDARD: Select and use appropriate statistical methods to analyze data
LEARNING EXPECTATIONS




Rework a set of data (represented by a table, scatterplot, stem-and-leaf diagram, box-and-whisker diagram, bar
graph, pie chart, and /or line graph) into one or more of the other forms.
Use appropriate statistics (e.g., mean, median, mode, and range) to communicate information about the data.
Explore how data distribution affects shape, center, and spread.
Generate a scatterplot, find a line of best fit, and use it to make predictions applied to authentic data. (Use both
paper-pencil and technology.)
NCTM LEARNING STANDARD: Develop and evaluate inferences and predictions that are based
on data
LEARNING EXPECTATIONS


Explore how sample size and population size may affect the validity of predictions from a set of data.
Explore designs of surveys, polls, and experiments to assess the validity of their results and to identify potential
sources of bias; identify the types of conclusions that can be drawn.
NCTM LEARNING STANDARD: Understand and apply basic concepts of probability
LEARNING EXPECTATIONS


Use area diagrams to determine the probability of events, e.g., dartboard or spinners.
Explore the relationship between the theoretical probability of simple events and the experimental outcome from
simulations.
MEASUREMENT
NCTM LEARNING STANDARD: Understand measurable attributes of objects and the units,
systems, and processes of measurement
LEARNING EXPECTATIONS

Relate changes in the measurement of one attribute of an object to changes in other attributes, e.g., how changing
the radius or height of a cylinder affects its surface area or volume.
NCTM LEARNING STANDARD: Apply appropriate techniques, tools, and formulas to determine
measurements
LEARNING EXPECTATIONS







Given the formula, find the surface area and volume of prisms, pyramids, cylinders, and cones.
Solve problems involving perimeter, circumference, area, lateral area, surface area, volume, angle measure, and
arc length, e.g., find the volume of a sphere with a specified circumference.
Apply similarity and the relationships of special triangles to the solution of problems such as indirect
measurement problems.
Use the Pythagorean Theorem and right triangle trigonometry to solve real world problems.
Given the formula, convert between metric and English units of measure.
Explore the scientific use of different systems of measurement.
Explore the effects of rounding on measurements and on computed values from measurements.
Mathematics For All
November 2, 2000
Page 52 of 55
GRADES 11-12
By the end of tenth grade, all students should understand the foundations of mathematical concepts and skills
that will prepare them for work, for further study, and to be contributing citizens. Perhaps as important,
though, is that students believe themselves to be capable and competent learners of mathematics. No young
person should be prevented from pursuing an interest by lack or perceived lack of mathematical understanding
and skills. The writers of Mathematics For All believe that no single course of study in grades 11 and 12 is
right for every child. We are well aware, though, that such thinking in the past has led to curriculum that sets
low expectations of most students while pushing others to calculus and beyond prematurely.
We have struggled to define exactly what constitutes learning standards and expectations for grades 11 and 12
and have not been able to achieve a satisfactory solution. We propose the following standards for all students
as a beginning point for discussion with our colleagues across the Commonwealth.
By the end of the twelfth grade, every student should:
• Understand number systems including matrices and complex numbers, and operate with very large and
very small numbers.
• Decide whether a problem needs an exact solution, an approximation or an estimate.
• Use algebra and geometry to describe, analyze and solve real world problems including projectile
motion and population growth.
• Be confident and flexible problem-solvers, bringing a wide range of skills and experience to each new
situation.
• Become increasingly sophisticated at reasoning and evaluating arguments.
• Use mathematical language, notation and representations to communicate with others precisely and
coherently about mathematics.
• Understand and critique an article that uses statistics in the form of percents, lists, charts or graphs.
All students should be encouraged to take four years of math to maximize their preparation for a wide
variety of technical, health, laboratory, and other skilled jobs as well as to pursue higher education.
Mathematics For All
November 2, 2000
Page 53 of 55
Appendix A: Criteria for Evaluating Instructional
Materials and Programs in Mathematics
Not at all
1. Student Experiences
Involve them in inquiry-based learning and problem
solving
Enable them to investigate important mathematical
concepts
Provide multiple pathways to develop concepts and
communicate ideas and solutions
Include use of manipulatives and tools to explore,
model, and analyze situations and communicate
findings
Foster collaboration and reflection
Attend to diverse cultural and economic backgrounds
Provide developmentally appropriate activities which
can accommodate the range of abilities and learning
styles found in classrooms
Draw on a variety of instructional resources (e.g.,
trade books, measuring tools, information technology,
manipulatives, primary sources, and electronic
networks)
Focus on current mathematical knowledge that is
accurately represented
Provide opportunities to ask their own questions and
conduct their own investigations
Include assessment prior knowledge, imbedded
assessments, and performance measures
Use real world ideas, topics, and contexts that are
appropriate and engaging for students
2. Mathematical Content
Reflects the learning expectations in Mathematics For
All
Is mathematically accurate and current
Uses real-world contexts
Provides opportunities for students to work as a
mathematician
Uses language and illustrations that are free of bias
and reflect the diversity of our society
Inadequately
Adequately
Strongly
Exceptionally
Mathematics For All
November 2, 2000
Page 54 of 55
Not at all
3. Organization and Structure
Provide for in-depth, inquiry-based investigations of
major mathematical concepts
Provide cohesive units that build conceptual
understanding
Emphasize connections within and across disciplines
Incorporate appropriate use of instructional
technology (computers, calculators)
Incorporate materials that are appropriate and
engaging for students
Include clear instructions on using tools, equipment,
and materials
Include a master source of materials and resources
4. Teacher Support Materials
Provide an overview of content
Provide suggestions to inform and engage parents and
other community members
Incorporate a variety of strategies to engage and
stimulate all students (open-ended questions, journals,
manipulatives, visual, auditory and kinesthetic
activities)
Provide a list of required instructional materials and
reference useful supporting materials (videos, trade
books, software, web sites, electronic networks)
Suggest ways for teachers to adapt the materials to
meet the needs of all students
Give background information to support various
learning approaches (cooperative groups, student as
teacher, independent research, learning centers, field
trips)
Include use of instructional technology to help
students visualize complex concepts, analyze and
refine information, and communicate solutions
Provide examples of student responses with rubrics to
evaluate the assessments
Inadequately
Adequately
Strongly
Exceptionally
Mathematics For All
November 2, 2000
Page 55 of 55
Not at all
5. Student Assessment Materials
Are free of racial, cultural, ethnic, linguistic, gender,
and physical bias
Align with student experiences
Are embedded in the instructional program, occurring
throughout the unit, not just at the end
Incorporate multiple forms of assessment (oral and
written work, student demonstrations, student selfassessment, projects, tests, quizzes, teacher
observations, individual and group assessments,
portfolios, and journals)
Focus on both the process (predicting, modeling,
making inferences, reasoning) and content of learning
Are useful to provide information about student
learning to inform the teacher’s instruction
6. Program Development and Implementation
Reflect current research on teaching and learning
Provide access to information regarding the evidence
of effectiveness
Provide published materials that include suggestions,
strategies, and models for successful implementation
at the classroom level
Provide published materials that include suggestions,
strategies, and models for successful implementation
at the school or district level
Inadequately
Adequately
Strongly
Exceptionally