Number Sense and Foundational Numeracy

What Is Number Sense?

Number sense is the flexible, intuitive understanding of numbers and their relationships. It includes knowing what numbers mean, how they compare to one another, and how operations like addition or multiplication change them. As a paraprofessional, your job is not to reteach a lesson from scratch but to support the classroom teacher's approach by reinforcing concepts at the student's current level of understanding. A strong grasp of number sense helps you recognize where a student is struggling and choose the right tool to help.

Place Value: The Foundation

Our number system is a base-ten positional system. Every digit in a number has a value based on its position. In the number 347, the 3 stands for 3 hundreds (300), the 4 stands for 4 tens (40), and the 7 stands for 7 ones (7). When students confuse digits and values—writing 34 when they mean 43, for example—they are signaling a place-value gap. Recognizing this pattern allows you to target support precisely.

Base-ten blocks make place value visible: a small cube represents one, a rod of ten cubes represents ten, and a flat of ten rods represents one hundred. Using these blocks, a student can hold 347 in their hands before they ever write it on paper.

The Concrete-Representational-Abstract (CRA) Sequence

The CRA sequence is a research-supported instructional framework widely used in elementary mathematics education. It moves students through three stages of understanding.

Concrete (C): Students use physical objects—manipulatives—to model a mathematical idea. Examples include base-ten blocks, counters, two-color chips, and number lines drawn on a desk. The hands-on stage builds the mental image that later stages depend on.

Representational (R): Students draw pictures or diagrams that stand for the physical objects. A student might draw ten-sticks and ones-dots instead of handling actual blocks, or sketch equal groups of circles to show multiplication. This stage bridges the gap between touching objects and working with symbols.

Abstract (A): Students work with numerals and operation symbols alone—the standard algorithm or equation. For example, writing 24 + 13 = 37 without any picture or block.

These stages are not strictly linear one-time events. A student may work at the abstract stage for addition but need the concrete stage again when division is introduced. As a paraprofessional, you observe which stage a student is currently at and meet them there.

Modeling the Four Operations with CRA

Addition: At the concrete stage, students combine two groups of blocks and count the total. At the representational stage, they draw the groups and cross-count. At the abstract stage, they apply the standard algorithm, carrying (regrouping) as needed.

Subtraction: Concrete—students remove blocks from a group and count what remains, or they use a number line to count back. Representational—students cross out drawn objects or mark jumps on a printed number line. Abstract—students apply the standard subtraction algorithm, borrowing (regrouping) from a higher place value when the digit being subtracted is larger.

Multiplication: Multiplication is repeated addition. At the concrete stage, 3 × 4 means three groups of four counters. Students build the groups physically and count all. At the representational stage, they draw three circles each containing four dots. At the abstract stage, they write 3 × 4 = 12.

Division: Division is repeated subtraction or equal sharing. At the concrete stage, 12 ÷ 3 means sharing 12 counters equally into 3 groups and finding that each group has 4. At the representational stage, students draw the groups. At the abstract stage, they write the equation or long-division format.

Using Manipulatives Safely and Effectively

Most manipulatives used in elementary classrooms are small objects. When supporting young learners, follow school policy on choking hazards—small counters and chips are not appropriate for children who put objects in their mouths. Always check your school's guidelines before distributing materials. Keep the work area organized so pieces do not become a tripping hazard on the floor. Store manipulatives in labeled containers after each session.

When working with students who have sensory sensitivities, be aware that some may find certain textures distracting. Offer alternatives (for example, drawn diagrams instead of physical chips) when a manipulative becomes a barrier rather than a bridge.

Your Role as a Paraprofessional

Your responsibility is to support—not replace—the classroom teacher's instruction. Before a small-group session, confirm with the teacher which CRA stage the group is working at and which manipulatives are approved for use. During the session, use guiding questions rather than giving answers: "What do you notice about your groups?" or "How many ones blocks do we have altogether?" After the session, report observations about student understanding back to the teacher so instruction can be adjusted.

The ParaPro Assessment mathematics section tests your ability to understand and apply basic mathematical concepts and to recognize how to assist students with those concepts. Knowing the CRA framework and being able to identify place value, addition, subtraction, multiplication, and division at each stage directly supports your performance on that exam and in daily practice.