Virtual manipulatives for mathematics

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In mathematics education, virtual manipulatives are a relatively new technology modeled after existing manipulatives such as base ten blocks, coins, blocks, tangrams, rulers, fraction bars, algebra tiles, geoboards, geometric plane, and solids figures. They are usually in the form of Java or Flash applets. Virtual manipulatives allow teachers to allow for efficient use of multiple representations and to provide concrete models of abstract mathematical concepts for learners of mathematics. Research suggests that students may also develop more connected understandings of mathematical concepts when they use virtual manipulatives (Moyer, Niezgoda, & Stanley, 2005).[1]

Many[who?] believe that virtual manipulatives can be particularly helpful to students with language difficulties, including English Language Learners (ELL). ELL students usually have trouble explaining what they are learning in mathematics classes. With virtual manipulatives, such students may be able to clarify their thoughts and demonstrate it to others in a much more effective way. For example, with base ten blocks, students may use the place-value layout to show their understanding.

Manipulatives by themselves have little meaning. It is important for teachers to make the mathematical meaning of manipulatives clear and help the students to build connections between the concrete materials and abstract symbols. Virtual manipulatives usually have this built-in structure. Many virtual manipulative activities give students hints and feedback with pop-ups and help features. More traditional concrete manipulatives are not conducive to comprehension without direct instructor assistance. For example, in using tangrams, students can practically copy a design made from pattern blocks. When a block is near a correct location, it will snap into place. This virtual manipulative includes a hint function that will show the correct location of all the blocks.

Although relatively new, virtual manipulatives can support learning mathematics for all students which includes those with learning disabilities and ELL learners. Virtual manipulatives can be included into the general academic curriculum and not just used as an extra student activity. If they are used wisely, virtual manipulatives can provide students with opportunities for guided discovery which can help them to build a better understanding of mathematical concepts and ultimately exhibit measurable learning skills.

To analyze the study and implementation of virtual manipulatives, a framework proposed the use of the well-known Concrete, Pictorial (also known as Representational) and Abstract cognitive learning levels (CPAor CRA) with the addition of the “Virtual” cognitive level (V), which is abbreviated as CPVA (Ortiz, 2017; Ortiz, Eisenreich, & Tapp, 2019). The “Virtual” cognitive level involves the use virtual manipulatives in the form of apps and applets (such as the ones presented in the “Notable collection of virtual manipulatives” below). It involves the virtual representation of the CPA cognitive levels in apps and applets. The “Virtual” level involves three digital-dynamic sublevels: virtual-Concrete, virtual-Pictorial and virtual-Abstract. This combination of cognitive levels provides an alternative way to analyze the development of curricular activities and research studies with a more consistent set of definitions.

Notable collections of virtual manipulatives[edit]

Wolfram Demonstrations Project

Wolfram Demonstrations Project contains around 11,000 Virtual manipulatives for math, science and engineering. They are provided in CDF format together with source code.

Didax Free Manipulatives Library

Didax is the U.S. branch of Philip & Tacey, Ltd of Hampshire, UK, who developed Unifix₢ Cubes in 1960, a popular math manipulative used throughout the world to teach counting and operations. Unifix cubes were created as a replacement for poppet beads, which rolled off student desks and were expensive to manufacture. The virtual manipulatives in this library are designed to be faithful to their physical counterparts and include minimal navigation or symbolic content.

Shodor Interactivate Activities

Shodor is a national resource for computational science education. They have offered online education tools such as Interactivate and the Computational Science Education Reference Desk (CSERD) since 1994. The activities are sorted from Grade 3 through Undergraduate.

National Library of Virtual Manipulatives

Utah State University has offered this collection of internet-based manipulatives since 1999. The activities are sorted from Pre-Kindergarten through High School. The manipulatives were originally developed in Java.

Illuminations: Activities

Illuminations has been found on a section of the website for the National Council of Teachers of Mathematics since 2000. Students and teachers from Pre-Kindergarten through High School can use these interactivities.

MSTE at the University of Illinois

According to their website, "Mathematics Materials for Tomorrow's Teachers (M2T2) are a set of mathematics modules created in the spring of 2000 by a team consisting of teachers, administrators, university researchers, mathematicians, graduate students, and members of the Illinois State Board of Education." They are five modules. Each module is connected to one of the goals for mathematics in the Illinois Learning Standards. The content is at a middle school level.


  • Moyer, P. S., Bolyard, J. J., & Spikell, M. A. (2000). What are virtual manipulatives? [Online]. Teaching Children Mathematics, 8(6), 372-377. Available: [1]
  • Moyer, P. S., Niezgoda, D., & Stanley, J. (2005). Young children's use of virtual manipulatives and other forms of mathematical representations. In W. J. Masalaski & P. C. Elliot (Eds.), Technology-Supported Mathematics Learning Environments (pp. 17–34). Reston, VA: National Council of Teachers of Mathematics.
  • Ortiz, Enrique (2017).Pre-service teachers’ ability to identify and implement cognitive levels in mathematics learning. Issues in the Undergraduate Mathematics Preparation of School Teachers (IUMPST): The Journal (Technology), 3, pp. 1–14. Retrieved from pdf:
  • Ortiz, Enrique, Eisenreich, Heidi & Tapp, Laura (2019). Physical and virtual manipulative framework conceptions of undergraduate pre-service teachers. International Journal for Mathematics Teaching and Learning, 20(1), 62-84. Retrieved from

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