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.
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/
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
Montessori, Virtual Manipulatives for Language Arts and
Mathematics [Online]. Accessed 4/22/01: http://www.phil.cmu.edu/~montessori/
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
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:
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
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
- MaxwellWorld supports the exploration of
electrostatics, leading up to the concept of Gauss'
- 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
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
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
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:
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.