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Introduction

Virtual math manipulatives are visual representations of concrete math manipulatives that are digitally accessed through a variety of websites or apps on digital devices including tablets, phones and computers.^[1] Virtual math manipulatives are modeled after concrete math manipulatives that are commonly used in classrooms to concretely represent abstract mathematical concepts and support student understanding of mathematical content. ^[2] ^[3]

Common manipulatives include: base ten blocks, coins, blocks, tangrams, rulers, fraction bars, algebra tiles, geoboards, geometric plane, and solids figures. Students can engage virtually/digitally with the manipulatives mimicking the use the of the concrete math manipulatives.

Notable collections of virtual manipulatives

Wolfram Demonstrations Project

http://demonstrations.wolfram.com/

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

http://www.didax.com/virtual-manipulatives-for-math

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

http://www.shodor.org/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

http://nlvm.usu.edu/

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

http://illuminations.nctm.org/Default.aspx

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

The Office for Mathematics, Science, and Technology Education (MSTE) at the University of Illinois at Urbana-Champaign has two good resources. The MSTE Online Resource Catalog has existed since 1994. Also, M2T2 includes an extensive List of Mathematics Applets.

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.

References

^ Moyer, P.S. (2002). "What are Virtual Manipulatives?". Teaching Children Mathematics. 8: 372–377.
^ Carbonneau, K.J. (2013). "A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives". Journal of Educational Psychology. 105: 380–400.
^ Silva R., Costa C., Martins, F. (2021). "Using Mathematical Modelling and Virtual Manipulatives to Teach Elementary Mathematics". Technology and Innovation in Learning, Teaching and Education. vol 1884 – via https://doi.org/10.1007/978-3-030-73988-1_6. {{cite journal}}: |volume= has extra text (help); External link in |via= (help)CS1 maint: multiple names: authors list (link)

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 http://www.k-12prep.math.ttu.edu/journal/3.technology/volume.shtml pdf: http://www.k-12prep.math.ttu.edu/journal/3.technology/ortiz01/article.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 https://www.cimt.org.uk/ijmtl/index.php/IJMTL/article/view/116

External links

[1] Moyer, P.S. (2002). "What are Virtual Manipulatives?". Teaching Children Mathematics. 8: 372–377.

[2] Carbonneau, K.J. (2013). "A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives". Journal of Educational Psychology. 105: 380–400.

[3] Silva R., Costa C., Martins, F. (2021). "Using Mathematical Modelling and Virtual Manipulatives to Teach Elementary Mathematics". Technology and Innovation in Learning, Teaching and Education. vol 1884 – via https://doi.org/10.1007/978-3-030-73988-1_6. {{cite journal}}: |volume= has extra text (help); External link in |via= (help)CS1 maint: multiple names: authors list (link)

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 {{advert|type=advocacy for this technology|date=December 2016}}
-In [[mathematics education]], '''virtual manipulatives''' are a relatively new technology modeled after existing [[manipulative (mathematics education)|manipulative]]s such as [[base ten blocks]], [[coins]], [[Toy block|blocks]], [[tangrams]], [[rulers]], [[fraction bars]], [[algebra tiles]], [[geoboard]]s, [[Plane (geometry)|geometric plane]], and [[geometric solid|solids figures]]. They are usually in the form of [[Java applet|Java]] or [[Adobe Flash|Flash]] applets. Virtual manipulatives allow teachers to allow for efficient use of [[Multiple representations (mathematics education)|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).<ref>[http://my.nctm.org/eresources/article_summary.asp?URI=TCM2002-02-372a&from=B What are Virtual Manipulatives?]</ref>
+=== Introduction ===
-Many{{who|date=December 2016}} 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.
+Virtual math manipulatives are visual representations of concrete math manipulatives that are digitally accessed through a variety of websites or apps on digital devices including tablets, phones and computers.<ref>{{Cite journal|last=Moyer|first=P.S.|date=2002|title=What are Virtual Manipulatives?|journal=Teaching Children Mathematics|volume=8|pages=372-377}}</ref> Virtual math manipulatives ''are modeled after concrete math manipulatives that are commonly used in classrooms'' to concretely represent abstract mathematical concepts and support student understanding of mathematical content. <ref>{{Cite journal|last=Carbonneau|first=K.J.|date=2013|title=A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives.|journal=Journal of Educational Psychology|volume=105|pages=380-400}}</ref> <ref>{{Cite journal|last=Silva R., Costa C., Martins|first=F.|date=2021|title=Using Mathematical Modelling and Virtual Manipulatives to Teach Elementary Mathematics.|url=https://doi.org/10.1007/978-3-030-73988-1_6.|journal=Technology and Innovation in Learning, Teaching and Education|volume=vol 1884|via=https://doi.org/10.1007/978-3-030-73988-1_6}}</ref>
+Common manipulatives include: '' [[base ten blocks]], [[coins]], [[Toy block|blocks]], [[tangrams]], [[rulers]], [[fraction bars]], [[algebra tiles]], [[Geoboard|geoboards]], [[Plane (geometry)|geometric plane]], and [[Geometric solid|solids figures]].'' Students can engage virtually/digitally with the manipulatives mimicking the use the of the concrete math manipulatives.
-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'' (''CPA''or ''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==