Last updated: 2007
Most children enter school confident in their own abilities; they are curious and eager to learn
more. They make sense of the world by reasoning and problem solving. Young students are
building beliefs about what mathematics is, about what it means to know and do mathematics,
and about themselves as mathematical learners. Students use mathematical tools, such as
manipulative materials and technology, to develop conceptual understanding and solve problems
as they do mathematics. Students, as mathematicians, learn best through participatory
experiences throughout the instruction of the mathematics curriculum.
Recognizing that no term captures completely all aspects of expertise, competence, knowledge,
and facility in mathematics, the term mathematical proficiency has been chosen to capture what
it means to learn mathematics successfully. Mathematical proficiency has five strands:
computing (carrying out mathematical procedures flexibly, accurately, efficiently, and
appropriately), understanding (comprehending mathematical concepts, operations, and relations),
applying (ability to formulate, represent, and solve mathematical problems), reasoning (logically
explaining and justifying a solution to a problem), and engaging (seeing mathematics as sensible,
useful, and doable, and being able to do the work) (NRC, 2001).
The most important observation about the five strands of mathematical proficiency is that they
are interwoven and interdependent. This observation has implications for how students acquire
mathematical proficiency, how teachers develop that proficiency in their students, and how
teachers are educated to achieve that goal. At any given moment during a mathematics lesson or
unit, one or two strands might be emphasized. But all the strands must eventually be addressed
so that the links among them are strengthened. The integrated and balanced development of all
five strands of mathematical proficiency should guide the teaching and learning of school mathematics. Instruction should not be based on the extreme positions that students learn solely
by internalizing what a teacher or book says, or solely by inventing mathematics on their own.
The Elementary Mathematics Core describes what students should know and be able to do at the
end of each of the K-6 grade levels. It was developed and revised by a community of Utah
mathematics teachers, mathematicians, university mathematics educators, and State Office of
Education specialists. It was critiqued by an advisory committee representing a wide variety of
people from the community, as well as an external review committee. The Core reflects the
current philosophy of mathematics education that is expressed in national documents developed
by the National Council of Teachers of Mathematics, the American Association for the
Advancement of Science, and the National Research Council. This Mathematics Core has the
endorsement of the Utah Council of Teachers of Mathematics. The Core reflects high standards
of achievement in mathematics for all students.
Guidelines Used in Developing the Elementary Mathematics Core
The Core is:
Consistent With the Nature of Learning
In the early grades, children are forming attitudes and habits for learning. It is important that
instruction maximizes students' potential and gives them understanding of the intertwined nature
of learning. The main intent of mathematics instruction is for students to value and use
mathematics as a process to understand the world. The Core is designed to produce an integrated
set of Intended Learning Outcomes for students.
The Core has been designed so that, wherever possible, the ideas taught within a particular grade
level have a logical and natural connection with each other and with those of earlier grades.
Efforts have also been made to select topics and skills that integrate well with one another and
with other subject areas appropriate to grade level. In addition, there is an upward articulation of
mathematical concepts and skills. This spiraling is intended to prepare students to understand
and use more complex mathematical concepts and skills as they advance through the learning
The Core takes into account the psychological and social readiness of students. It builds from
concrete experiences to more abstract understandings. The Core focuses on providing
experiences with concepts that students can explore and understand in depth to build the
foundation for future mathematical learning experiences.
Reflective of Successful Teaching Practices
Learning through play, movement, and adventure is critical to the early development of the mind
and body. The Core emphasizes student exploration. The Core is designed to encourage a
variety of interactive learning opportunities. Instruction should include recognition of the role of
mathematics in the classroom, school, and community.
By emphasizing depth rather than breadth, the Elementary Mathematics Core seeks to empower
students by providing a comprehensive background in mathematics. Teachers are expected to
teach all the standards and objectives specified in the Core for their grade level, but may add
related concepts and skills.
Teachers and others who are familiar with Utah students, classrooms, teachers, and schools have
designed the Core. It can be taught with easily obtained resources and materials. A handbook is
also available for teachers and has sample lessons on each topic for each grade level. The
handbook is a document that will grow as teachers add exemplary lessons aligned with the new
Useful and Relevant
This curriculum relates directly to student needs and interests. The relevance of mathematics to
other endeavors enables students to transfer skills gained from mathematics instruction into their
other school subjects and into their lives outside the classroom.
Reliant Upon Effective Assessment Practices
Student achievement of the standards and objectives in this Core is best assessed using a variety
of assessment instruments. Performance tests are particularly appropriate to evaluate student
mastery of mathematical processes and problem-solving skills. Teachers should use a variety of
classroom assessment approaches in conjunction with standard assessment instruments to inform
instruction. Sample test items, keyed to each Core Standard, may be located on the "Utah
Mathematics Home Page" at http://www.schools.utah.gov/curr/Math/elem/default.htm. Observation of students
engaged in instructional activities is highly recommended as a way to assess students' skills as
well as attitudes toward learning. The nature of the questions posed by students provides
important evidence of their understanding of mathematics.
Based Upon the National Council of Teachers of Mathematics Curriculum Focal Points
In 2006, the National Council of Teachers of Mathematics (NCTM) published Curriculum Focal
Points for Prekindergarten through Grade 8 Mathematics (NCTM, 2006). This document is
available online at http://www.nctm.org/focalpoints. This document describes three focal points
for each grade level. NCTM's focal points are areas of emphasis recommended for the
curriculum of each grade level. The focal points within a grade are not the entire curriculum for
that particular grade; however, Utah's Core Curriculum was designed to include these areas of
Organization of the Elementary Mathematics Core
The Core is designed to help teachers organize and deliver instruction.
- Each grade level begins with a brief description of areas of instructional emphasis
which can serve as organizing structures for curriculum design and instruction.
- The INTENDED LEARNING OUTCOMES (ILOs) describe the skills and attitudes
students should acquire as a result of successful mathematics instruction. They are
found at the beginning of each grade level and are an integral part of the Core.
- A STANDARD is a broad statement of what students are expected to understand.
Several Objectives are listed under each Standard.
- An OBJECTIVE is a more focused description of what students need to know and be
able to do at the completion of instruction. If students have mastered the Objectives
associated with a given Standard, they have mastered that Standard at that grade
level. Several Indicators are described for each Objective.
- INDICATORS are observable or measurable student actions that enable students to
master an Objective. Indicators can help guide classroom instruction.
- MATHEMATICAL LANGUAGE AND SYMBOLS STUDENTS SHOULD USE
includes language and symbols students should use in oral and written language.
- EXPLORATORY CONCEPTS AND SKILLS are included to establish connections
with learning in subsequent grade levels. They are not intended to be assessed at the
grade level indicated.