C-P-A + Technology = C-V and V-P-A  Approach 

As technology becomes more prevalent in our society, teachers have the opportunity to present an external representation of virtual manipulatives. Virtual manipulatives are a type of technological tool that provides virtual representation which can be manipulated dynamically to develop mathematical conceptual understanding. This is unlike a pictorial representation, which is static. Virtual manipulatives serve to marry the cognitive gap between the concrete and pictorial representations, into what researchers have defined as the C-V and V-P-A two-part approach (Lee and Tan, 2014). 

Benefits: 

• Engagement 

  • Motivation - Since students are exposed to digital resources and multimedia in their everyday lives, they are comfortable and enthusiastic about the use of virtual manipulatives in the learning process (Lee and Tan, 2014). 

• Effectiveness 

  • Personalization - Virtual manipulatives encourage creativity as students manipulate "on-screen objects to test hypotheses and experiment ideas" (Lee and Ferrucci, 2012). It promotes student advocacy as students have fun constructing meaning with various mathematics concepts. 

  • Differentiation - Virtual manipulatives allow students to produce multiple examples or skip elements on the applet/application based on individualized needs.

  • Learning Together - Virtual manipulatives stimulate interactivity between the learner and mathematics. It has integrated support for students to share and build on each others' ideas to develop understanding. 

  • Multiple representations - Students can problem-solve and use multiple forms of representation, especially through the efficient coupling of visual representation with other forms of representation 

• Efficiency 

  • Classroom management - Virtual manipulatives can be accessed quickly so teachers can free up logistical time distributing and retrieving concrete manipulatives.

  • Unlimited supply - Virtual manipulatives are available to all students as long as they have access to a device. Physical manipulatives are limited to what is available/how much is available in a classroom. 

  • Unlimited access - Virtual manipulatives are available 24/7, even for students to access at home. 

An Example: The role of virtual manipulatives on the Concrete-Pictorial-Abstract approach in teaching primary mathematics (Lee and Tan, 2014)

In Lee and Tan's case study (2014), students in Teacher X's classroom were seen to be more comfortable and fluid with the use of virtual manipulatives compared to concrete manipulatives. This was because virtual manipulatives enabled the creation of quick, precise mathematical representations through specific features (i.e. automatically size or snap together fraction pieces) that supported students' initial ideas about the connections between representations. In addition, students were quicker to engage themselves in higher-order thinking because they were able to test out their own hypotheses and experiment with ideas, which enhanced their creativity and thinking overall. 

An Example: Learning Mathematics with Technology: The Influence of Virtual Manipulatives on Different Achievement Groups (Moyer-Packenham and Suh, 2012)

Moyer-Packenham and Suh's case study (2012) examined how virtual manipulatives (a fraction applet) affected different achievement groups of fifth-graders during a teaching experiment on fraction equivalence and fraction addition with unlike denominators. 

High-progress students were able to use virtual manipulatives combined with mental math strategies, pattern identification, and recognizing equivalence and proportional relationships to complete mathematical tasks. Middle-progress students used more step-by-step methodical processes and relied on pictorial models in the fraction applet rather than proportional relationships. Low-progress learners relied heavily on counting strategies and pictorial models in the fraction applet to complete mathematical tasks. They also used trial and error with the pictorial model because the virtual manipulatives had a constraint-support structure which provided the opportunity to just focus on the connections between the actions of the two systems (notation and visuals) (Moyer-Packenham and Suh, 2012). Virtual manipulatives scaffolded the learning process by disallowing incorrect moves. The overall results demonstrated that there was a significant gain for all students who used virtual manipulatives between the pre-and post-tests; however, low-progress learners benefitted the most and made the most statistically significant gains. 

A potential equity issue to note is how high-progress students versus low-progress students will respond to the use of virtual manipulatives and exposure to technological tools beyond curriculum time.