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Virtual Manipulatives

Computerized versions of "concrete" hands-on manipulatives.

Manipulatives (hands-on materials) are commonly used in a variety of subjects and grade levels of instruction. In recent years, a number of websites have been developed that contain "virtual" versions of some of these manipulatives. The references given below provide information about some of these websites. Click here for the Virtual Manipulatives page on the Oregon Technology in Education Council Website.

References

Clements, D. H. (1999). 'Concrete' manipulatives, concrete ideas. Contemporary Issues in Early Childhood, 1(1), 45-60 [Online]. Accessed 3/22/01: http://www.gse.buffalo.edu/org/buildingblocks/
NewsLetters/Concrete_Yelland.htm.

The Web reference is for a slightly updated version of the original article. The discussion covers both physical manipulatives and computer-generated manipulatives. The article suggests that it is not inherently obvious that one form of manipulative is better than another, and that computer-generated manipulatives may well be superior in some cases.

Hartshorn, Robert and Boren, Sue (1990). Experiential learning of mathematics: Using manipulatives. ERIC Digest. Accessed 11/4/03: http://www.ericfacility.net/databases/ERIC_Digests/ed321967.html. Quoting from the Website:

Experiential education is based on the idea that active involvement enhances students' learning. Applying this idea to mathematics is difficult, in part, because mathematics is so "abstract." One practical route for bringing experience to bear on students' mathematical understanding, however, is the use of manipulatives. Teachers in the primary grades have generally accepted the importance of manipulatives. Moreover, recent studies of students' learning of mathematical concepts and processes have created new interest in the use of manipulatives across all grades.

In this Digest "manipulatives" will be understood to refer to objects that can be touched and moved by students to introduce or reinforce a mathematical concept. The following discussion examines recent research about the use of manipulatives. It also speculates on some of the challenges that will affect their use in the future.

Both Pestalozzi, in the 19th century, and Montessori, in the early 20th century, advocated the active involvement of children in the learning process. In every decade since 1940, the National Council of Teachers of Mathematics (NCTM) has encouraged the use of manipulatives at all grade levels. Every recent issue of the "Arithmetic Teacher" has described uses of manipulatives. In fact, the entire February 1986 issue considered answers to the practical questions of why, when, what, how, and with whom manipulative materials should be used.

Research suggests that manipulatives are particularly useful in helping children move from the concrete to the abstract level. Teachers, however, must choose activities and manipulatives carefully to support the introduction of abstract symbols. Heddens divided the transitional iconic level (the level between concrete and abstract) further into the semiconcrete and semiabstract levels, in the following way:

The semiconcrete level is a representation of a real situation; pictures of the real items are used rather than the items themselves. The semiabstract level involves a symbolic representation of concrete items, but the pictures do not look like the objects for which they stand. (Heddens, 1986, p.14)

Howden (1986) places specific manipulatives on this continuum. These manipulatives rank from the concrete to the abstract. In place value, for example (going from concrete to abstract), they include pebbles, bundled straws, base-ten blocks, chip-trading, and the abacus. Howden cautions that building the bridge between the concrete and abstract levels requires careful attention. She notes that, even if children can solve a given problem at the concrete level, they may not be able to solve the same problem at the abstract level. This problem occurs if the bridge has not been structured by a careful choice of manipulatives.

Suydam and Higgins (1977), in a review of activity-based mathematics learning in grades K-8, determined that mathematics achievement increased when manipulatives were used. Sowell (1989) performed a meta-analysis of 60 studies to examine the effectiveness of various types of manipulatives with kindergarten through postsecondary students. Although these studies indicate that manipulatives can be effective, they suggest that manipulatives have not been used by many teachers.

Math Learning Center (MLC): Virtual Math Manipulatives on the Web. Accessed 7/21/01: http://www.mlc.pdx.edu/mathlinks.html.

Contains links to a number of websites that provide free virtual manipulatives useful in mathematics education.

Montessori, Virtual Manipulatives for Language Arts and Mathematics [Online]. Accessed 4/22/01: http://www.phil.cmu.edu/~montessori/
guest/guest.html. Quoting from the Website:

Homewood Montessori, located in Pittsburgh, provides a unique teaching method based on the principles of Maria Montessori. One of the main teaching tools used at this school are manipulatives. Manipulatives are physical representations of abstract concepts that allow children to interact and build a solid knowledge base. These materials are used for both mathematics and the language arts.

Virtual manipulatives are online versions of these materials. These virtual manipulatives introduce children to computers and allow for many students to use the materials at once.

National Library of Virtual Manipulatives for Interactive Mathematics. Accessed 4/1/04: http://matti.usu.edu/nlvm/nav/index.html. Quoting from the Website:

This is a three-year NSF supported project to develop a library of uniquely interactive, web-based virtual manipulatives or concept tutorials, mostly in the form of Java applets, for mathematics instruction (K-8 emphasis). The project includes dissemination and extensive internal and external evaluation.

Learning and understanding mathematics, at every level, requires student engagement. Mathematics is not, as has been said, a spectator sport. Too much of current instruction fails to actively involve students. One way to address the problem is through the use of manipulatives, physical objects that help students visualize relationships and applications. We can now use computers to create virtual learning environments to address the same goals.

NCTM (n.d.). Principles & Standards: Electronic Examples Accessed 2/16/06: http://standards.nctm.org/document/eexamples/#6-8.

This site provides interactive figures for various grade levels. These are designed to help in explanation and exploration of various principles in the standards.

Resnick, M. et al. (1998). Digital Manipulatives: New Toys to Think With [Online]. Accessed 4/22/01: http://llk.media.mit.edu/papers/1998/dig-manip/. Quoting from the paper:

In many educational settings, manipulative materials (such as Cuisenaire Rods and Pattern Blocks) play an important role in children's learning, enabling children to explore mathematical and scientific concepts (such as number and shape) through direct manipulation of physical objects. Our group at the MIT Media Lab has developed a new generation of "digital manipulatives" -- computationally-enhanced versions of traditional children's toys. These new manipulatives enable children to explore a new set of concepts (in particular, "systems concepts" such as feedback and emergence) that have previously been considered "too advanced" for children to learn. In this paper, we discuss four of our digital manipulatives -- computationally-augmented versions of blocks, beads, balls, and badges.

ScienceSpace. Accessed 3/17/03: http://www.virtual.gmu.edu/. Quoting from the Website:

The purpose of Project ScienceSpace is to explore the strengths and limits of virtual reality (sensory immersion, 3-D representation) as a medium for science education. This project is a joint research venture among George Mason University, the University of Houston, and NASA's Johnson Space Center. Dr. Chris Dede from George Mason University is the project Co-Principal Investigator and has developed this web site; Dr. R. Bowen Loftin of the University of Houston is the Principal investigator.

Project ScienceSpace is a collection of immersive virtual worlds designed to aid students in mastering challenging concepts in science. ScienceSpace now consists of three worlds:

  • NewtonWorld provides an environment for investigating the kinematics and dynamics of one-dimensional motion.
  • MaxwellWorld supports the exploration of electrostatics, leading up to the concept of Gauss' Law.
  • PaulingWorld enables the study of molecular structures via a variety of representations.

Spicer, Judy (April 13, 2000). Virtual Manipulatives: A New Tool for Hands-On Math [Online]. Accessed 4/22/01: http://www.enc.org/resources/freestuff/focus/
equity/documents/0,1948,FOC-001754-index,00.shtm. Quoting from the article:

ENC's virtual manipulatives presentation at the National Council of Teachers of Mathematics convention in Chicago began with a question and ended with a collective "Wow!"

Here is the question:

Why use the Internet for instructional purposes in the middle school mathematics classroom?

One answer is that the Internet enables the learner to see and explore concepts not readily accessible in other mediums. For example, virtual manipulatives offer computer-generated objects that can be manipulated by a computer user. Virtual manipulatives have the power to make visible that which is hard to see--and impossible to imagine.

During the presentation, we demonstrated sites that make it possible to interactively explore the relationship between the equation of a line and its slope, to see an image of a fourth dimension hypercube, and to begin to get a feel for infinity. The visual beauty of the mathematics found at these sites created excitement in the audience, and we believe, will wow even the most uninterested students.

Virtual Manipulatives: A New Tool for Hands-on Math [Online]. Accessed 4/22/01: http://www.negaresa.org/n200020.html.

This Website is maintained by the Northeast Georgia Regional Educational Service Agency. It addresses and brief descriptions of nine websites that contain virtual manipulatives for use in mathematics education.

Virtual Manipulatives [Online]. Accessed 4/22/01: http://www.matti.usu.edu/nlvm/

The Website has materials for K-12.] Quoting from the Website:
This is a three-year project to develop a library of uniquely interactive, web-based virtual manipulatives or concept tutorials, mostly in the form of Java applets, for mathematics instruction (K-8 emphasis). The project includes dissemination and extensive internal and external evaluation.

Learning and understanding mathematics, at every level, requires student engagement. Mathematics is not, as has been said, a spectator sport. Too much of current instruction fails to actively involve students. One way to address the problem is through the use of manipulatives, physical objects that help students visualize relationships and applications. We can now use computers to create virtual learning environments to address the same goals.

Prior to this time, there has been very little done to create good computer-based mathematical manipulatives or learning tools at elementary and middle school levels with any degree of interactivity. Our Utah State University team is building Java-based mathematical tools and editors that allow us to create exciting new approaches to interactive mathematical instruction. The use of Java as a programming language provides platform independence and web-based accessibility.

Ultimately we will make all materials available at several sources on the Internet, creating a national library from which teachers may freely draw to enrich their mathematics classrooms. The materials will also be of importance for the mathematical training of both in-service and pre-service elementary teachers.