What Are Virtual Manipulatives?

1 what Are Virtual manipulatives ? s a result of innovations in technology, the prevalence of the Internet, and the increasing avail- A ability of computers in classrooms and homes, an enhanced approach for teaching and learn- ing mathematics using manipulatives and computers is emerging. This new approach essen- tially creates a new class of manipulatives , called Virtual manipulatives , as well as new capabilities, or toolkits, for computer programs that use visual representations. These new Virtual manipulatives have all the useful properties of existing computer manipulatives while overcoming many of their disadvantages, yet very little is known or written about them. The purpose of this article is to establish a working definition of Virtual manipulatives , highlight examples of Virtual manipulatives on the Internet, and discuss their current and potential classroom use. ing a name and definition that uniquely describe Defining Virtual these dynamic images is equally important to avoid manipulatives confusion.

2 Two types of representations on the World Wide Static visual representations are essentially pic- Web are being called Virtual manipulatives these tures. They are the sorts of visual images ordinar- are static and dynamic visual representations of ily associated with pictures in books, drawings on concrete manipulatives (Spicer 2000). Conse- an overhead projector, sketches on a chalkboard, quently, two types of Virtual manipulatives and so on. Although such representations resemble are available as teaching and learning tools. concrete manipulatives , they cannot be used in the Patricia S. Moyer, Because one of these types is far more same ways that concrete manipulatives can. That Johnna J. Bolyard, and powerful and has much greater utility and is, a student can actually slide, flip, and turn con- Mark A. Spikell potential for teaching, drawing the distinc- crete manipulatives but cannot perform the same tion between the two is important. Establish- actions with a static picture of the concrete manip- ulative.

3 These static visual representations are not true Virtual manipulatives . Patricia Moyer, and Mark Spikell, are colleagues at In contrast, dynamic visual representations of George Mason University (GMU), Fairfax, VA 22030. Moyer conducts research on teachers' concrete manipulatives are essentially objects.. and students' uses of representation in mathematics. Spikell is the director of the Metamedia They are visual images on the computer that are Mathematics Program and coordinator of the Educational Leadership Programs in Mathe- just like pictures in books, drawings on an over- matics and Science in the Graduate School of Education at GMU. Johnna Bolyard, jjbolyard teaches eighth-grade mathematics at Farmwell Station Middle School, Ash- head projector, sketches on a chalkboard, and so burn, VA 20147. She is currently a doctoral student in the Metamedia Mathematics Program on. In addition, these dynamic visual representa- in the Graduate School of Education at GMU.

4 Tions can be manipulated in the same ways that a concrete manipulative can. Just as a student can 372 TEACHING CHILDREN MATHEMATICS. Copyright 2002 The National Council of Teachers of Mathematics, Inc. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM. slide, flip, and turn a concrete manipulative by who would provide interactive opportunities for hand, he or she can use a computer mouse to actu- their students to construct mathematical knowl- ally slide, flip, and turn the dynamic visual repre- edge must learn to distinguish between what is sentation as if it were a three-dimensional object. truly a Virtual manipulative and what is merely This kind of visual representation is truly a Virtual Virtual .. manipulative. Although many people currently use the term Static images or representations Virtual manipulative to refer to any computer- Virtual sites consist of static images or representa- generated image that appears on a monitor and is tions.

5 An example of such a site is Visualize Frac- intended to represent concrete manipulatives , this tions at This site pre- meaning is much too simplistic to be descriptive or sents a number line and circular representations of useful. The ability to manipulate the visual repre- fractions; users answer questions using the proper sentation, or object, on the computer connects the symbolic notation. The images change on the user with the real teaching and learning power of screen to prompt responses, but the user cannot Virtual manipulatives , that is, the opportunity to move or manipulate the image. The site does not make meaning and see relationships as a result of offer interactive opportunities for users to construct one's own actions. This kind of user engagement meaning on their own. distinguishes Virtual manipulative sites from those sites where the act of pointing and clicking results Computer-manipulated images or in the computer's providing an answer in visual or representations symbolic form.

6 We think that the term Virtual Some sites present a more representational view manipulative should be restricted to describing this but still do not allow users to manipulate images. interactive capability. For example, the computer might move the images In our view, a Virtual manipulative is best in response to a command from defined as an interactive, Web-based visual repre- the user, but the user does not sentation of a dynamic object that presents oppor- directly manipulate the object. tunities for constructing mathematical knowledge. Again, because these sites pre- Students can slide, Currently, Virtual manipulatives are modeled on clude user involvement, they the concrete manipulatives commonly used in cannot be considered to incorpo- flip, and turn a visual schools, such as pattern blocks, tangrams, fraction rate true Virtual manipulatives . representation as bars, geoboards, and geometric solids. When such One of these sites is Room 108. objects are available through the Web, they may be at ~kingskid if it were a three- considered Virtual .

7 However, their ability to be / , which pro- used interactively that is, to allow the user to vides an applet for multipli- dimensional object engage and control the physical actions of these cation in its Math section. objects combined with the opportunities that they Numbers are arranged in a mul- offer to discover and construct mathematical prin- tiplication table, and users click ciples and relationships, distinguishes them as vir- on two factors and the equals sign ( , 2 3 =); the tual manipulatives . With current technology, the computer then displays the answer as both a 2 3. most common way of effecting these movements rectangular array of squares and as the numeral 6. on a monitor is by using a computer mouse to The program does not allow the user to take an point, click, and drag. The future will undoubtedly active role in creating either representation. The user bring many other ways of moving Virtual manipu- cannot actually move the object and, through that latives, perhaps through such features as voice movement, create mathematical meaning.

8 Commands and infrared signals. Virtual manipulative Web sites Web sites that incorporate Virtual manipulatives Virtual manipulatives offer much more interactive experiences for users. and the Merely Virtual Manipula Math with JAVA ( As previously mentioned, distinct differences in / ), for example, allows users in the interactivity exist among Web sites that use Virtual middle school section to choose from several manipulatives . On the one hand are sites that limit applets that demonstrate and verify the Pytha- user involvement by displaying mathematical con- gorean theorem. In these demonstrations, the user cepts as static images on the monitor. On the other is presented with a right triangle to adjust. The the- hand are dynamic sites that let users extensively orem is investigated by moving pieces of the manipulate and even alter the representations to squares that represent a2 and b2 and placing them explore concepts and construct meaning. Teachers into the area that represents c2.

9 The various applets FEBRUARY 2002 373. use different methods for fitting the pieces into c2. tives ( ). This site, Another site that offers a similar experience is the funded by the National Science Foundation, fea- Pythagoras' Theorem page at tures Web-based Virtual manipulatives or concept .com/sketchpad/java_gsp/PIT. HTM. tutorials, primarily in the form of Java applets, for David E. Joyce's The Geometry Applet, mathematics instruction, mainly at the elementary version ( ~djoyce/java and middle school levels ( / ) and Paul Flavin's Java /nav- ). Users can choose from 3d Viewer ( ~imaging/java- the following strands: numbers and operations, ) offer dynamic experiences in algebra, geometry, measurement, and data analysis three-dimensional geometry. Texas A & M Univer- and probability. The site includes instructions on sity's two Algebra Tiles pages ( using the manipulatives , activities to accompany .edu/~strader/Mathematics/Algebra/Algebr aTiles the site, and links to the NCTM's Standards and ) also (NCTM 2000).

10 Researchers developing this site offer users more dynamic experiences for exploring have several important goals, including interactiv- concepts in algebra. Here, users arrange the tiles in ity for the user, the user's control of variable a rectangle with dimensions that represent the fac- aspects, and the opportunity to explore and dis- tors of a given quadratic expression. cover mathematical principles and relationships Dynamic sites that provide physical manipula- (Cannon, Heal, and Wellman 2000). tives in Virtual form for teachers and students to Another site for which various types of manip- control are true Virtual manipulative sites. Along ulative tools are being developed is George with the Virtual toolkits, these sites offer descrip- Mason University's Dr. Super's Virtual Math tions of their potential uses for mathematics manipulatives Project ( instruction and practical instructions on how to /~drsuper/). The goal of this project is to create manipulate the Virtual and develop curriculum materials and technology- forms.