Effective Mathematics Interventions

By Margaret M. Flores, PhD, BCBA-D, Auburn University
Co-author of Making Mathematics Accessible for Elementary Students Who Struggle: Using CRA/CSA Interventions


According to the National Center for Educational Statistics (2016), the 2015 National Assessment of Educational Progress showed that 18% of fourth grade students performed below basic levels of achievement, meaning that they did not demonstrate mastery of fundamental skills. Students’ mathematical difficulties begin with understanding numbers, basic operations and their novice conceptions lead to further difficulties with complex operations and fractions (Fuchs et al.; Jordan & Hanich, 2003; 2016). Students who struggle in mathematics comprise a diverse group which includes students with identified disabilities as well as students without disabilities (Powell, Fuchs, & Fuchs, 2013). There is a critical need for effective implementation of interventions that have been shown to be effective through research. One effective approach that can be adapted across mathematical concepts is the concrete-representational/semi-concrete-abstract sequence (Miller, Stringfellow, Kaffar, & Mancl, 2011; Witzel, Furguson, & Mink, 2012; CRA/CSA).

What Is CRA/CSA?

The CRA/CSA sequence in an instructional approach to mathematics that emphasizes conceptual understanding prior to procedural knowledge and fluency. There are three phases: concrete, representational/semi-concrete, and abstract. The concrete phase of instruction involves the use of objects to complete mathematical tasks or solve problems. During this phase, teachers explicitly teach concepts through the manipulation of objects. The representational/semi-concrete phase continues to focus on the development of conceptual understanding, but problems are solved using pictures and student-made drawings. Once students demonstrate understanding of the target mathematics concept at the representational/semi-concrete levels, they learn to solve problems using just numbers, the abstract phase. During the abstract phase, the focus of instruction is on procedural knowledge and fluency. The benefit of including the CRA/CSA sequence into mathematics interventions is that the concrete and representational/semi-concrete phases provide students with needed remediation in their understanding of whole numbers, the base ten system, operations, and rational numbers (fractions). The physical manipulation of objects, drawing, and visual aid of pictures fill in the gaps that exist in their prerequisite knowledge and understanding about mathematics. Another benefit of these physical and visual aids is that they assist students in making meaning of mathematical language and using language to explain their computation or problem solving.

CRA/CSA and Number Concepts

The CRA/CSA sequence has been shown to be effective in teaching young children and elementary students number concepts. Researchers used CRA/CSA to teach preschool students, with and without disabilities, counting skills. This included number sense in the form of visual counting or recognizing that four objects were represented by the numeral four without physically touching the objects (Hinton, Flores, Schweck, & Burton, 2015; Hinton, Flores, & Strozier, 2015). Elementary students also successfully learned how to count this way using CRA/CSA. In addition, Hinton and Flores (submitted) taught rounding skills using CRA/CSA. Using base ten blocks and drawings representing base ten blocks, students learned how to round numbers to the nearest ten and hundred. After abstract instruction using just numbers, students quickly and accurately completed rounding tasks. Mercer and Miller (1992) taught place value to elementary students with and without disabilities using CRA/CSA.

CRA/CSA and Basic Operations

Miller and Mercer (1992) and Mercer and Miller (1992) taught elementary students, with and without disabilities, basic operations using the CRA/CSA sequence. This included addition, subtraction, multiplication, and division. Using objects and drawings, students learned the conceptual meaning of each operation: addition is combining, subtraction is separating, multiplication is combining of groups that are the same size, and division is the separation of groups that are the same size. After instruction at the concrete and representational/semi-concrete phases, students learned a simple strategy to assist in computation using just numbers. This set of steps served as a reminder to (a) attend to the numbers and the operational sign, (b) remember that problems can be drawn if the student has not memorized the fact, and (c) write the answer. Students who participated in this large study become fluent in basic operations and their accuracy in computation increased significantly.

CRA/CSA Complex Operations

Researchers also used the CRA/CSA sequence to teach regrouping skills associated with addition, subtraction, and multiplication (Miller & Kaffar, 2011; Mancl, Miller, & Kennedy, 2012; Flores, 2011; Flores & Hinton, in press; Flores, Hinton, & Strozier, 2014; Flores, Schweck, & Hinton, 2014; Flores & Franklin, 2014). Difficulties faced by students within each of these studies were related to poor conceptions of numbers and the base ten system. The concrete and representational/semi-concrete phases of instruction involved the use of base ten blocks and drawings that bolstered students’ understanding of numbers and why regrouping is necessary in when adding and subtracting large numbers. These studies included students with and without disabilities and led to significant gains in accuracy and fluency.

CRA/CSA and Fractions

CRA/CSA has been shown as an effective way to teach rational numbers or fraction concepts (Butler, Miller, Crehan, Babbit, & Pierce, 2003; Flores & Hinton, submitted). Butler et al. studies the necessity of including a concrete phase within instruction. Students successfully leaned to make equivalent fractions, but those who used fraction blocks prior to drawings performed better than those who only received instruction using drawings. Flores and Hinton taught elementary students equivalency using CRA/CSA as well as comparison of fractions to decimals. At the concrete phase, students made fractions using fraction blocks as well as sets of objects. At the representational level, students shaded shapes and marked number lines. In both studies, concrete and representational/semi-concrete instruction allowed students to understand the proportional nature of fractions which led to their mastery of more complex concepts such as equivalence and relations to decimals.


The CRA/CSA sequence has been shown to be effective across a variety of elementary mathematics concepts. The materials needed are simple; base ten blocks and counters are readily available in elementary schools. However, it may be difficult for teachers to implement and replicate the research as journal articles are not written in ways that provide detailed descriptions of each lesson component. Therefore, in order to close the gap in mathematical achievement, there is a need for more user-friendly guides for implementation of the CRA/CSA sequence.


Butler, F. M., Miller, S. P., Crehan, K., Babbitt, B., & Pierce, T. (2003). Fraction instruction for students with mathematics disabilities. Learning Disabilities Research and Practice, 18, 99–111.

Flores, M. M., Hinton, V. M., & Strozier, S. D. (2014). Teaching subtraction and multiplication with regrouping using the concrete-representational-abstract sequence and strategic instruction model. Learning Disabilities Research and Practice, 29, 75–88.

Flores, M. M., & Franklin, T. M. (2014). Teaching multiplication with regrouping using the concrete-representational-abstract sequence and the strategic instruction model. Journal of American Special Education Professionals, 6, 133–148.

Flores, M. M., Schweck, K. B., & Hinton, V. M. (2014). Teaching multiplication with regrouping to students with learning disabilities. Learning Disabilities Research & Practice, 29(4), 171–183.

Fuchs, L. S., Schumacher, R. F., Long, J., Namkung, J., Malone, A., Wang, A., Hamlett, C. L., Jordan, N. C., Siegler, R. S., & Changas, P. (2016). Effects of intervention to improve at-risk fourth graders’ understanding, calculations, and word problems with fractions. Elementary School Journal, 116(4), 625–651.

Jordan, N. C., & Hanich, L. B. (2003). Characteristics of children with moderate mathematics deficiencies: A longitudinal perspective. Learning Disabilities Research & Practice, 18, 213–221. doi:10.1111/1540-5826.00076

Mancl, D. B., Miller, S. P., & Kennedy, M. (2012). Using the concrete-representational-abstract sequence with integrated strategy instruction to teach subtraction with regrouping to students with learning disabilities. Learning Disabilities Research and Practice, 27(4), 152–166.

Mercer, C. D., & Miller, S. P. (1992). Teaching students with learning problems in math to acquire, understand, and apply basic math facts. Remedial and Special Education, 13(3), 19-35. doi: 10.1177/074193259201300303

Miller, S. P., & Kaffar, B. J. (2011). Developing addition with regrouping competence among second grade students with mathematics difficulties. Investigations in Mathematics Learning, 4(1), 24–49.

Miller, S. P., & Mercer, C. (1992). CSA: Acquiring and retaining math skills. Intervention in School and Clinic, 28(2), 105–110.

Miller, S. P., Stringfellow, J. L., Kaffar, B. J., & Mancl, D. B. (2011). Developing computation competence among students who struggle with mathematics. Teaching Exceptional Children, 44(2), 38–44.

Witzel, B. S., Furguson, C. J., & Mink, D. V. (2012). Number sense: Strategies for helping preschool through grade three children develop math skills. Young Children 89–94


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