# Mathematics

The Mathematics content area of the Colorado Academic Standards contains four standards that apply to children in kindergarten through third grade: Number Sense, Properties, and Operations; Algebra and Functions; Data Analysis, Statistics, and Probability; and Shape, Dimension, and Geometric Relationships. In addition, a set of mathematical practice standards describes the critical thinking processes and proficiencies that children develop throughout the course of their mathematics education. These standards are adapted from the Common Core State Standards for Mathematics. This section provides a broad overview of the requirements of each of these standards in grades K–3 and explains how the content in these Guidelines at earlier ages prepares children for meeting these standards in their formal schooling.

Mathematical Practice

The overarching Mathematical Practice standards include problem solving, constructing arguments and evaluating the reasoning of others, looking for and using structure, and looking for and expressing regularity in repeated reasoning. In preparation for these skills in grades K–3, preschoolers develop the ability to understand and solve simple problems by seeking solutions and understanding cause and effect relationships as described in the Logic and Reasoning: Reasoning and Problem-Solving subdomain of the Guidelines for children ages 3–5 years. The Mathematical Practice standards also include abstract and quantitative reasoning and modeling. In preparation for these skills in grades K–3, preschoolers develop the ability to use symbols or objects to represent something else, as described in the Logic and Reasoning: Symbolic Representation domain of the Guidelines for children ages 3–5years.

Number Sense, Properties, and Operations

The first standard in the Mathematics content area focuses on children’s understanding of quantity, ways of representing numbers, relationships among numbers, and number systems. Children learn that numbers are governed by properties, and understanding these properties leads to fluency with operations. Precursor skills in the Guidelines for children ages 3–5 years are found in the Number Concepts and Quantities subdomain and the Operations and Algebraic Thinking subdomain. As described in those sections, preschool age children are learning that numbers represent quantities and have ordinal properties. They also use numbers to describe relationships and solve problems. Children build on this knowledge in kindergarten through third grade. Kindergarten children learn to represent, relate, and operate on whole numbers. Children in first grade develop understanding of addition, subtraction, and strategies for addition and subtraction; whole number relationships; and place value, including grouping in tens and ones. Second grade children extend their understanding of base ten notation and build fluency with addition and subtraction. Third grade children develop understanding of multiplication, division, and strategies for multiplication and division within 100, and fractions, especially unit fractions (i.e., fractions with numerator 1).

Algebra and Functions

The second standard address how children understand and use numbers. This standard is supported by the skills and understandings addressed under the first standard for Number Sense. Precursor skills in the Guidelines for children age 3–5 years are found in the Number Concepts and Quantities subdomain and the Operations and Algebraic Thinking subdomain. As described in those sections, preschool age children are learning that numbers represent quantities and have ordinal properties. They also use numbers to describe relationships and solve problems. Children build on this knowledge in kindergarten through third grade. Children’s understanding of arithmetic in early grades supports their developing understanding of equations and functions in later grades. In kindergarten, children begin to model and represent addition and subtraction problems with objects, fingers, and drawings. In first and second grade, children use more complex problem solving strategies and solve more complex problems involving two- and three-digit numbers. In third grade, children learn to represent and solve multiplication and division problems and apply strategies to other problems. The mathematics addressed in this standard is important for real-world applications that require modeling, expression, and equations.

Data, Statistics, and Probability

The third standard provides children with tools to understand information and uncertainty. Children ask questions and gather and use data to answer their questions. For preschool and kindergarten children, the basic mathematics skills needed to understand this standard are integrated into the other standards—there are no specific guidelines for Data Analysis, Statistics, and Probability. Beginning in first grade, children know that visual displays of information can be used to answer questions. Children in second and third grade also know that data can be displayed and described in a variety of formats.

Shape, Dimension, and Geometric Relationships

The final standard in the Mathematics content area focuses on children’s comprehension of space and shape. Precursor skills in the Guidelines for children ages 3–5 years are found in the Geometry and Spatial Sense subdomain and the Measurement and Data subdomain. Preschool age children recognize, describe, and compare shapes, including their position in space. They also compare and order objects by their length, weight, and size. As children progress into kindergarten, they continue to focus on describing shapes and space. First grade children learn about linear measurement; measuring lengths as iterating length units; and reasoning about attributes of, and composing and decomposing, geometric shapes. In second grade, children learn to use standard units of measure and describe and analyze shapes. Children in third grade develop understanding of the structure of rectangular arrays and of area, and learn to describe and analyze two-dimensional shapes.