Studies have evaluated the effectiveness of manipulatives as a tool in mathematics instruction. One line of research has studied the Concrete-Representational-Abstract (CRA) sequence. This form of explicit instruction moves students from concrete manipulatives to pictorial representations of those manipulatives, and from pictorial representations to abstract concepts. Butler, Miller, Crehan, Babbitt, and Pierce (2003) compared the effectiveness of teaching fraction concepts to students with learning disabilities using a CRA approach versus a Representational-Abstract (RA) approach (which starts with pictorial representations and then moves to abstract concepts without using concrete manipulatives). Although both groups improved in terms of their understanding of fractions, the CRA group had higher scores overall than the RA group.

A study by Witzel, Mercer, and Miller (2003) also supports the effectiveness of adopting a CRA approach to develop the basic mathematics skills of students with learning disabilities. Students were taught to solve algebraic equations using either a CRA approach or a traditional approach. The study involved 34 matched pairs of students in Grades 6 and 7 who had either been diagnosed with learning disabilities or categorized as at risk for learning problems. Although both groups showed improvement after a four-week intervention, the CRA group significantly outperformed the group that had received traditional instruction.

In another CRA study (Maccini & Hughes, 2000), six adolescents with learning disabilities used algebra tiles to represent algebra word problems during the concrete phase of instruction. The students were able to transition successfully to pictorial and ultimately symbolic representations of the problems.

With the rapid development of technology, virtual manipulatives have become widely used in mathematics instruction and have been shown to have a positive impact on students’ mathematics achievements. Bolyard and Moyer-Packenham (2012) conducted a quasi-experimental study to examine the effectiveness of virtual manipulatives in helping sixth-grade students make sense of integer arithmetic. The results revealed that the use of virtual manipulatives helped students improve integer computation achievement.

Reimer and Moyer (2005) investigated the performance of 19 third-grade students during a two-week unit on fractions that used virtual manipulatives. More than half of the students improved their conceptual understanding of fractions on a teacher-designed measure. In another study of 19 second-grade students, Moyer, Niezgoda, and Stanley (2005) observed that virtual Base-10 Blocks enabled students to demonstrate more sophisticated strategies and explanations of place value. Bolyard and Moyer-Packenham (2006) studied the use of virtual manipulatives with 99 sixth-grade students learning addition and subtraction of integers. The students showed significant gains in achievement, and the researchers concluded that virtual manipulatives can support learning these concepts.

Bouck, Satsangi, Doughty, and Courtney (2014) used virtual manipulatives and concrete manipulatives to teach subtraction skills to a group of elementary-aged students with autism spectrum disorder. Although both types of manipulatives helped students with mathematics problem solving, students using virtual manipulatives achieved greater accuracy and faster independence in solving the subtracion problems. Suh and Moyer (2007) compared the use of concrete and virtual manipulatives among third-grade students studying algebraic thinking. Both types of manipulatives were associated with higher achievement and increased flexibility in representing algebraic concepts. Steen, Brooks, and Lyon (2006) compared the academic achievement of a group of first-grade students who used virtual manipulatives for practice in geometry instruction (the treatment group) with another group who did not use virtual manipulatives (the control group). A total of 31 students were randomly assigned to either the treatment or control group. Achievement was measured by the Grade 1 and Grade 2 assessments provided by the classroom textbook’s publisher. The treatment group improved significantly on both the Grade 1 and Grade 2 tests, while the control group showed significant improvement only on the Grade 1 test. The teacher of the treatment group also noted that her students showed increased motivation and spent more time on task.

Teachers also play an important role in helping students understand the concepts that manipulatives represent (Bjorklund, 2014; Moyer, 2001). This was highlighted in a one-year study of 10 middle school mathematics teachers and their use of manipulatives (Moyer, 2001). Teachers who were unable to represent mathematics concepts were more likely to use manipulatives as a diversionary rather than an instructional activity.

The CRA studies described above also demonstrate the importance of structure and guidance in linking concrete materials with abstract concepts. For example, students in Maccini and Hughes’ study (2000) were taught not only to use algebra tiles to represent word problems, but also to use a structured strategy in solving them. In the Reimer and Moyer study (2005), students benefited from another important aspect of guidance: feedback. When students were interviewed regarding their impressions of the virtual manipulatives, an emergent theme was their appreciation of the immediate feedback provided by computer-based manipulatives. In addition, it is important for teachers to motivate students intrinsically when using manipulatives in mathematics instruction in order to achieve the desired outcomes (Jones, Uribe-Fiorez, & Wilkins, 2011).