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Integrating IT Into Each Subject Area

Virtual Manipulatives

IT-based manipulatives are an important supplement to and/or replacement for certain types of hands-on "concrete" manipulatives often used in math education.

Manipulatives (blocks, rods, bean sticks, etc.) are commonly used in mathematics education. 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.

It appears that there may be a fine dividing line between a concrete manipulative and a virtual manipulative. For example, does it make any difference in student learning whether a physical spinner is used to generate data, or whether a computerized spinner (that may well look the same) is used?

One of the most important general developments in science in recent years has been the idea of Computational Science. Thus, we now have Computational Biology, Computational Chemistry, etc. (The same idea applies to other disciplines, and these other disciplines have had varying levels of success in incorporating IT within their basic fabric.) And, as might be expected, Computational Mathematics is now an important branch of mathematics.

The essence of Computational "XXXX" (name a discipline) is the development and use of mathematical models that can be implemented on a computer. Thus we have 1) The "real thing."; The mathematical model.; and 3) The Computational Model (simulation). This has deep educational ramifications, both in math education and in each other discipline. In brief summary:

  1. Math education needs to include both math modeling and computer implementation of math models (computer simulations). This needs to be taught/learned in a manner that transfers to disciplines outside of math.
  2. Each discipline needs to be examined from the point of view of the math modeling and computer simulation that has become and/or is becoming an integrated component of the discipline. Thus, each teacher of a specific discipline needs to help students learn current and potential roles of math modeling and computer simulation to represent and help solve the problems of the discipline.

As an example, suppose that a math teacher is teaching students to use a spreadsheet. The spreadsheet is an excellent aid to developing math models of certain types of business problem situations. And, of course, spreadsheet models/simulations are used in many other disciplines. Thus, the math teacher is responsible both for teaching use of the spreadsheet to develop and implement models, but also how to use this tool in a variety of areas such as business, social studies, and so on. The business teacher, the social students teacher, and so on have the responsibility both of having students develop an appropriate level of fluency in modeling/simulation within their disciplines, but also carrying students to greater depths in the types of problems that are addressed by this approach.

In summary, when a math teacher has children use a virtual manipulative, the math teacher has the opportunity to help students learn some of the underlying ideas of math modeling and computer simulation. This is quite a different goal than merely using the virtual manipulative as a substitute (be it inferior of superior) the the real concrete manipulative that it models.

References

Clements, D. H. (1999). 'Concrete' manipulatives, concrete ideas. Contemporary Issues in Early Childhood, 1(1), 45-60 [Online]. Accessed 10/26/05: 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.

CT4ME. Computing Technology for Math Excellence. Accessed 3/24/06: http://www.ct4me.net/math_manipulatives.htm. Quoting from the Website:

Math Manipulatives, part of the Math Web Resources and Standardized Test Preparation series, contains resources that enable students to interact online.  We begin with a short essay on virtual manipulatives.  We provide a definition of a virtual manipulative, discuss the the role of virtual manipulatives in the classroom and provide cautionary statements about using and overusing manipulatives.  Or, you may jump right to the resources for Manipulatives on the Web. PDA, and calculator resources are also included. 

In "What are Virtual Manipulatives?," Patricia Moyer, Johnna Bolyard, and Mark Spikell (2002) define a virtual manipulative as "an interactive, Web-based visual representation of a dynamic object that presents opportunities for constructing mathematical knowledge" (p. 373).  Static and dynamic virtual models can be found on the Web, but static models are not true virtual manipulatives.  Static models look like physical concrete manipulatives that have traditionally been used in classrooms, but they are essentially pictures and learners cannot actually manipulate them.  "...[U]ser engagement distinguishes virtual manipulative sites from those sites where the act of pointing and clicking results in the computer's providing an answer in visual or symbolic form" (p. 373).  The key is for students to be able to construct meaning on their own by using the mouse to control physical actions of objects by sliding, flipping, turning, and rotating them. 

Hartshorn, Robert and Boren, Sue (1990). Experiential Learning of Mathematics: Using Manipulatives. ERIC Digest. [Online]. Accessed 1/25/02: http://www.ed.gov/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.

[Mention is made of one researcher who pays special attention to the scale or pathway from Concrete to Abstract. A Virtual Manipulative may lie someplace between these extremes.]

Math Forum. Using Manipulatives. Accessed 1/30/05: http://mathforum.org/t2t/faq/faq.manipulatives.html. This is a set of quesitons and answers about math manipulatives. For example, quoting form the Website:

What role can manipulatives play in the classroom?

Individual students learn in different ways. When manipulatives are used, the senses are brought into learning: students can touch and move objects to make visual representations of mathematical concepts. Manipulatives can be used to represent both numbers and operations on those numbers. In addition to meeting the needs of students who learn best in this way, manipulatives afford the teacher new ways of visiting a topic.

Math Learning Center (MLC): Virtulative Math Manipulatives on the Wen [Online]. 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.

Moyer, Patricia, Johnna Bolyard, and Mark Spikell (February, 2003). What are Virtual Manipulatives? Teaching Children Mathematics. Accessed 1/30/05: http://my.nctm.org/eresources/view_media.asp?article_id=1902

This article establishes a definition for virtual manipulatives, provides examples of virtual manipulative Web sites, and discusses their potential uses for teaching mathematics in K-12 classrooms.

National Library for Virtual Manipulatives for Interative Mathematics [Online]. Accessed 10/25/05: http://matti.usu.edu/nlvm/nav/index.html. Quoting from the Website:

This is an NSF supported project that began in 1999 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-12 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.

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.

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.

Using Manipulatives (n.d.). Accessed 10/26/05: http://mathforum.org/t2t/faq/faq.manipulatives.html.

This is a piece of the Teacher2Teacher Website (http://mathforum.org/t2t/). It includes links to a variety of useful articles.

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.

 

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