Teaching subtraction with regrouping (borrowing) using a concrete-representational-abstract sequence

iStock_000013133930XSmallNew Ways to Teach Subtraction

Along with learning to tie one’s own shoes, understanding how to “regroup” or borrow in subtraction is one of the many triumphs of childhood.

Researchers have demonstrated that the use of manipulatives such as blocks and of visual representations is an effective way to teach addition, subtraction, multiplication and division to children struggling with math.

A small study recently published in Preventing School Failure demonstrates that an instructional sequence using manipulatives, then drawings may help students with disabilities learn the complex task of regrouping.

The author used a concrete-representational-abstract (CRA) instructional sequence with 6 3rd-grade students who had specific learning disabilities and students identified as at risk for failing math.

Students were identified by a multidisciplinary team. “The CRA instructional sequence provided the students with a scaffold from conceptual understanding to procedural knowledge in which the students became fluent.”

A Strategic Approach

The researcher, Margaret Flores, provided instruction 3 days per week for 30 minutes each day during the study. She described CRA and its rationale to students and obtained a commitment from the student in the form of a contract. Teacher and students agreed to work rigorously using strategies to master subtraction. Learning sheets for the lessons were divided into 3 sections: model problems, guided practice problems and independent practice problems.

Lessons 1-3 (Concrete). After establishing baseline for all students regarding subtraction with regrouping in the tens place, the researcher conducted the lessons with manipulative objects, specifically Base-10 blocks made out of foam.

Lessons 4-6 (Representational). After 3 lessons with 80% accuracy or better, the researcher used drawings of the Base-10 blocks with tens represented by vertical bars and ones by a horizontal line with tallies for each one. Borrowing was represented by circling one of the vertical bars and adding 10 tallies to the horizontal bar.

Lesson 7. (Transitional) To help them during the transition from concrete and representational to the abstract, students are taught a mnemonic strategy, DRAW:

  • Discover the sign
  • Read the problem
  • Answer or draw and check
  • Write the answer.

Lesson 8-10 (Abstract). After students could recite the DRAW strategy and solve problems with 80% accuracy, they moved to Lessons 8-10. Students were encouraged to answer problems from memory rather than using drawings, but they also could use the DRAW strategy.

After Lesson 10, instruction involved timed fluency activities. The criterion for phase change was 20 digits correct on a 2-minute probe across 3 consecutive probes, the author writes. The probes had 30 subtraction problems (2-digit numbers minus another 2-digit number) that required regrouping in the tens place.


The criterion of 20 digits is the norm for students finishing 2nd grade, she wrote. Of the participants, 5 of 6 maintained performance after 4 weeks of no instruction or practice opportunities.

“The students physically demonstrated their understanding of the regrouping procedures rather than memorizing steps,” the author writes. “After students’ understanding of the procedures were firm (demonstrated across three lessons), it was necessary to fade their dependence on manipulative objects. To accomplish this, the students drew pictures to represent the problem; last, I asked the students to follow the procedures without drawing.

“Teaching Subtraction With Regrouping to Students Experiencing Difficulty in Mathematics,” by Margaret Flores, Preventing School Failure, 2009, Volume 53, Number 3, pp. 145-152.

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