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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:
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. ReferencesClements, 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/ 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:
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. 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?
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/ 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. 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
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:
NCTM (n.d.). Principles & Standards: Electronic Examples Accessed 2/16/06: http://standards.nctm.org/
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/ 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!" Using Manipulatives (n.d.). Accessed 10/26/05: http://mathforum.org/t2t/faq/faq.manipulatives.html.
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.
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