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Number Sense includes both the conceptual understanding of whole and rational numbers and their magnitude, and the ability to represent and operate with these numbers. Students with good Number Sense can readily transition between representations of quantity in the real world and mathematical expressions, and therefore can easily access and use their number knowledge to inform their math problem solving.

Number Sense includes multiple skills that begin to develop in the early years (see the Math PK-2 model) and continue to be foundational to successful math learning and performance:

**Symbolic Number:**An understanding of how a quantity maps onto the symbols - or*numerals*- that represent it and the ability to combine those numerals;**Decomposition:**An understanding of how to break numbers into smaller parts;**Place Value:**An understanding of the base-10 system and the value of individual numerals in a multi-digit number; and**Estimation:**The ability to give a rough approximation of the magnitude of a number or value of a calculation.

While Number Sense includes understanding of the magnitude represented by both abstract symbols (i.e., numerals) and Non-symbolic Number representations (e.g., dots), research has shown that Symbolic Number knowledge is a stronger predictor of students' math outcomes.

Providing math tasks with high cognitive demand conveys high expectations for all students by challenging them to engage in higher-order thinking.

CRA is a sequential instructional approach during which students move from working with concrete materials to creating representational drawings to using abstract symbols.

Students activate more cognitive processes by exploring and representing their understandings in visual form.

Continual use of foundational skills with different problems reinforces a conceptual understanding of math skills.

Daily review strengthens previous learning and can lead to fluent recall.

Knowing the language of math is critical because students must use this language to understand math concepts and determine calculations needed.

Thinking of and about patterns encourages learners to look for and understand the rules and relationships that are critical components of mathematical reasoning.

Teaching students to recognize common problem structures helps them transfer solution methods from familiar to unfamiliar problems.

Discussing strategies for solving mathematics problems after initially letting students attempt to problem solve on their own helps them understand how to organize their mathematical thinking and intentionally tackle problems.

Analyzing incorrect worked examples is especially beneficial for helping students develop a conceptual understanding of mathematical processes.

In guided inquiry, teachers help students use their own language for constructing knowledge by active listening and questioning.

Spending time with new content helps move concepts and ideas into Long-term Memory.

Practicing until achieving several error-free attempts is critical for retention.

Math centers support learner interests and promote the development of more complex math skills and social interactions.

Math games allow students to practice many math skills in a fun, applied context.

Rhyming, alliteration, and other sound devices reinforce math skills development by activating the mental processes that promote memory.

When students have meaningful conversations about math and use math vocabulary, they develop the thinking, questioning, and explanation skills needed to master mathematical concepts.

By talking through their thinking at each step of a process, teachers can model what learning looks like.

Providing physical and virtual representations of numbers and math concepts helps activate mental processes.

Easy access to seeing the relationships between numbers promotes Number Sense as students see these connections repeatedly.

Visual representations help students understand what a number represents as well as recognize relationships between numbers.

Having students teach their knowledge, skills, and understanding to their classmates strengthens learning.

When teachers connect math to the students' world, students see how math is relevant and applicable to their daily lives.

Math games and manipulatives for vision differences support math development for learners with visual needs.

Children's literature can be a welcoming way to help students learn math vocabulary and concepts.

When students create their own number and word problems, they connect math concepts to their background knowledge and lived experiences.

Untimed tests provide students the opportunity to flexibly and productively work with numbers, further developing their problem-solving abilities.

Analyzing and discussing solved problems helps students develop a deeper understanding of abstract mathematical processes.

Writing that encourages students to articulate their understanding of math concepts or explain math ideas helps deepen students' mathematical understanding.

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