One of the telltale signs that a child is struggling with basic arithmetic is his or her use of counting strategies to do simple addition problems that are typically used by younger children.
Arithmetic interventions for children with math disabilities (MD) typically focus on memory retrieval with drill and practice, one of the more advanced skills students use in becoming proficient in number combinations (NCs) or math facts (additions and subtractions). But a recent study in Exceptional Children says that working with students on the more basic skill of counting strategies is a simple and effective intervention to use in math remediation.
“Due to the ease of implementation and the lower costs of the counting strategies approach to remediation, we relied on this approach to address participants’ NC deficits as we moved toward word-problem remediation,” write Lynn Fuchs and his team of researchers. Their study ” (A Framework for Remediating Number Combination Deficits) recommends that educators go beyond drill and practice in arithmetic remediation and develop math interventions based on counting and other skills children must use to be successful.
The team conducted 4 studies of interventions for children with math disabilities using different versions of software they designed called Math Flash. Not satisfied with results from the first study, which focused on drill and practice, researchers conducted studies of other interventions that placed more emphasis on the conceptual underpinnings of number combinations.
The arc of arithmetic skills
To become proficient in arithmetic, children rely on the following 3 skills, according to the study: counting strategies, decomposition strategies, and memory retrieval.
When children realize that the sum of 5 +1 is the number that occurs next, this provides the developmental scaffold for more abstract, sophisticated counting strategies, the authors write. They then discover the efficiency of counting from the first addend. So in the sum 2 + 3, for example, they begin counting from the smaller addend that occurs first (the max strategy). Eventually, they discover the most efficient strategy of counting not from the number that occurs first but from the largest number (min strategy)
As conceptual knowledge about numbers becomes more sophisticated, children learn that a hole can be decomposed into parts in different ways (e.g., 2 + 3=[2 + 2=4] +1=5). “As increasingly efficient counting and decomposition strategies help students consistently and quickly pair problems with correct answers in working memory, associations become established in long-term memory, and children gradually favor memory-based retrieval of answers.”
Children use a mix of strategies in number combinations, with counting and decomposition serving as back-ups for memory-based retrieval, the authors write. Students with math disabilities fail to make the shift to memory-based retrieval because they have a compromised number sense and greater difficulty with counting. When they do retrieve answers from memory, they commit more errors and are not as fast as their younger, typically developing counterparts, the researchers write. By the end of 3rd grade, most children should be adept at basic arithmetic, the researchers write. For children who are still struggling at this stage, educators should refer to these 3 skills or strategies as a remediation framework when designing or selecting interventions.
In conducting their studies with Math Flash, the researchers found that all the interventions that worked on the basic 3 skills had a similar impact on learning. But the counting interventions were the easiest to administer because they required less training and supervision of tutors, the researchers say. All interventions included a focus on number concepts.
In the counting strategies study , tutors used manipulatives and the number line to work with students. Central to the intervention, the researchers say, was that students were told if they didn’t immediately know the answer to an NC , they should “count up.”
Students were taught to count up using a number line made up of sets of 10 squares and eventually fingers. Subsequent lessons worked with sets of 5 (i.e., addition problems with 5 as the sum or subtraction problems with 5 as the minuend) progressing to sets of 6 and 7 and so on through 18. Each of the 48 lessons included work with computerized drill and flash cards warm-up. In the warm-up, tutors showed flash cards, one at a time, placing correct cards in a pile and instructing students to “count up” when they gave incorrect answers on cards.
Special education teachers should select or develop an intervention they believe will be most productive for a student, the authors say. Students who are nonresponsive to any of these interventions should receive more intensive remediation that integrates the 3 intervention approaches within a skills-based diagnostic-prescriptive scheme that is individualized for the student.
The 4th study in the series examined 2 counting strategies with and without deliberate practice. The counting strategies remediation with deliberate practice resulted in superior learning compared with counting instruction strategies instruction without deliberate practice, the researchers report. The deliberate practice was substantially less than in interventions focused on drill and practice and on decomposition strategies.
“This suggests the power of the counting strategies as a focus of remediation for helping students with MD derive solutions to NCs in a fluent and accurate manner, ” the authors write.
The 4 studies were conducted in different years with different groups of 3rd graders in Houston and Nashville.. Under the study entry criteria, students had to perform below the 26th percentile on the Wide Range Achievement Test ,(WRAT; Wilkinson, 1993) Arithmetic.
“A Framework for Remediating Number Combination Deficits,” by Lynn Fuchs et al., Exceptional Children, Volume 76, Number 2, pp. 135-156.