Concrete-representational-abstract (CRA) for math goals

By the IEP Atlas editorial teamLast updated November 1, 2026

Short answer

Concrete-representational-abstract (CRA) teaches a math skill in three stages: with objects the student moves, then with drawings, then with numbers and symbols only. Plan all three stages before you start, set a rule for moving on (such as 4 of 5 correct with an explanation), bridge each stage to the next, and write the stages into the goal's short-term objectives.

CRA is a sequence, not a set of manipulatives. Base-ten blocks in a bin are not CRA. CRA is the plan that moves a student from blocks to drawings to the written method, with clear points where you decide to move on or go back. Done that way, it gives a math goal built-in benchmarks.

This page walks through planning and running CRA, with one fictional student's fraction data from start to finish. For the wider set of math goals, see the math IEP goals pillar.

What CRA is and why it helps

Concrete: the student solves with objects: blocks, fraction strips, counters, algebra tiles.
Representational: the student solves with drawings that stand for the objects: tally marks, bar models, sketches of blocks.
Abstract: the student solves with numbers and symbols only.

Federal guidance on elementary math intervention recommends visual representations, including concrete objects and drawings, and explicit instruction that links them to symbols.^[1] The fractions practice guide recommends that students use models such as fraction strips and number lines to understand fraction size and operations before relying on rules.^[2] CRA is a practical way to sequence that advice.

What you need before you start

One target skill, stated narrowly (adding fractions with like denominators, not "fractions").
A matched set of materials for each stage.
A 5-problem probe you can give at every stage.
A written rule for moving on and for going back.
An accuracy data sheet with a column for the stage.

Step-by-step: planning and running CRA

Step 1: Pick one skill and plan all three stages

Write the concrete, representational and abstract version of the same problem before you teach anything. If you cannot draw the representational step, the bridge to symbols will be weak.

Skill	Concrete	Representational	Abstract
Two-digit subtraction with regrouping	Base-ten blocks on a mat	Sticks and dots, crossing out a stick	Written algorithm
Equivalent fractions	Fraction strips lined up	Shaded bars split into parts	2/3 = 4/6 by multiplying
Multiplication	Counters in equal groups	Arrays drawn on grid paper	4 times 6 = 24
One-step equations	Algebra tiles on a balance mat	Drawn tiles, crossing out pairs	x plus 3 = 7, so x = 4
Integer addition	Two-color counters	Drawn plus and minus circles	negative 5 plus 3 = negative 2

Step 2: Teach the concrete stage explicitly

Model with the objects while you think aloud. "I need to take away 7 ones, but I only have 2. I trade a ten for 10 ones." Then solve together, then let the student solve alone. Have the student say what each move means. Moving objects without words does not build understanding.

Step 3: Set the rule for moving on

A common rule: the student solves 4 of 5 problems correctly at this stage and can explain one in their own words, on 2 sessions in a row. Write the rule down so every adult uses the same one.

Step 4: Bridge to the representational stage

For the first few lessons, the student solves with objects and draws what they did right beside it. Then drawings alone. Keep drawings fast and simple: a stick for a ten, a dot for a one, a box split into parts for a fraction.

Step 5: Bridge to the abstract stage

Write the numbers and symbols beside each drawing, step by step. Point out where each mark in the drawing shows up in the written work. Then fade the drawing. Some students keep a quick sketch as a check for months, and that is fine.

Step 6: Go back a stage when errors show a concept gap

If a student in the abstract stage makes the same concept error 2 sessions in a row, return to the drawings or the objects for a lesson or two. Going back is part of the plan, not a failure. Repeated short probes graphed over time tell you when to move and when to return.^[3]

Step 7: Write CRA into the goal and objectives

The annual goal names the abstract stage. The short-term objectives name the stages on the way, which gives the team natural benchmarks for progress reports.

Worked example: CRA for adding fractions

Noor (fictional) is in grade 5. On 5 problems adding fractions with like denominators (such as 2/8 plus 3/8), she scores 1, 1 and 2 of 5 across three baseline probes. She adds both numerators and denominators: 2/8 plus 3/8 = 5/16.

Annual goal. Given 10 addition and subtraction problems with like denominators of 2, 3, 4, 6 and 8, Noor will write the correct sum or difference for 8 of 10 on 3 consecutive weekly probes.

Short-term objectives:

Given fraction strips, Noor will model and solve 4 of 5 problems and explain why the denominator stays the same.
Given shaded bar drawings, Noor will solve 4 of 5 problems.
Given problems written in symbols only, Noor will solve 8 of 10.

Week	Stage	Score	Explained?	Decision
Baseline	Abstract	1 of 5 (median)	No	Start concrete
1	Concrete, fraction strips	3 of 5	Partly	Stay
2	Concrete	4 of 5	Yes	Stay, rule needs 2 sessions
2	Concrete	5 of 5	Yes	Move to drawings
3	Representational	3 of 5	Yes	Stay
4	Representational	4 of 5, then 4 of 5	Yes	Move to symbols with drawings beside
5	Abstract with drawing	7 of 10	Yes	Fade drawing
6	Abstract	6 of 10	No, adding denominators again	Go back to drawings for 2 lessons
7	Abstract	8 of 10	Yes	Stay
8 to 9	Abstract	8, then 9 of 10	Yes	Goal met on 3 consecutive probes

The week 6 dip was a return of the old error. Two lessons with drawings fixed it. Without the CRA plan, the team might have read week 6 as the student "losing" the skill.

Common mistakes and better choices

Weak: Using manipulatives once as a fun lesson, then going straight to worksheets. Better: Plan all three stages and a rule for moving on.
Weak: Staying in the concrete stage for months. Better: Move on when the student meets the rule, then return briefly if needed.
Weak: Drawings that take longer than the objects. Better: Quick sketches: sticks, dots and split bars.
Weak: Jumping from objects to symbols. Better: Bridge with drawings beside the written steps.
Weak: A goal that says "will use manipulatives." Better: A goal at the abstract stage, with objectives for each earlier stage.
Weak: Treating a return to objects as regression. Better: Plan for it and record the stage on the data sheet.

Tools and templates

Base-ten blocks, fraction strips, two-color counters, algebra tiles, grid paper.
Math IEP goals for templates at each stage, such as modeling regrouping with base-ten blocks.
Regrouping IEP goals for a full CRA sequence on two-digit subtraction.
Teaching number sense with manipulatives for the earliest concrete work.
Schema-based instruction for word problems, which uses diagrams as its representational stage.
The goal checker to test a goal with CRA objectives.

Frequently asked questions

What does CRA stand for in math?

Concrete, representational, abstract. Students first solve with objects, then with drawings that stand for the objects, then with numbers and symbols only. The teacher plans when to move between stages and links each stage to the next.

How long should a student stay in each CRA stage?

Until the student meets a set rule, such as 4 of 5 correct with an explanation on 2 sessions in a row. For many skills that is a few lessons per stage. Some students need longer, and some return briefly to an earlier stage later.

Is CRA only for elementary students?

No. Algebra tiles for equations, two-color counters for integers and fraction strips for older students all follow CRA. The sequence works whenever a student follows a rule without understanding what it means.

Use it in IEP Atlas

Sources

Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades, What Works Clearinghouse, Institute of Education Sciences
Developing Effective Fractions Instruction for Kindergarten Through 8th Grade, What Works Clearinghouse, Institute of Education Sciences
Progress Monitoring: Data & Goal Setting, National Center on Intensive Intervention

Written and checked by the IEP Atlas editorial team under our editorial policy. This guide describes general practice and federal rules. It is not legal advice; your state may add requirements, and the IEP team makes the decisions for each student.