If you teach math, chances are you've given computer-based manipulatives a try to reinforce your students' understanding of key concepts. But are you confident you?re getting the most out of these dynamic tools for math learning? This month?s Class Ideas takes a look at virtual manipulatives through the lens of the new book Teaching Mathematics with Virtual Manipulatives by educator Patricia Moyer-Packenham. This comprehensive teacher guide lists criteria for the selection and use of virtual manipulatives, summarizes the current research on their effectiveness, and offers 15 sample lessons linked to the National Library of Virtual Manipulatives website.
We hope you enjoy our peek into this valuable new resource guide and this very relevant topic!
Virtual manipulatives are an exciting new technology for teaching mathematics. Unlike static pictures on the computer, virtual manipulatives are dynamic and flexible. These objects can be moved, rotated, flipped, and turned, and some can even be changed entirely.
Many of the virtual manipulatives currently available were designed based on physical manipulatives such as pattern blocks, tangrams, and other commercially produced geometric shapes and solids. Other virtual manipulatives were developed in the computer environment with no physical counterpart. Whether based on a physical manipulative or developed for use on a computer, the flexibility of virtual manipulatives makes them uniquely suited for teaching mathematics.
Unique Features and Capabilities of Virtual Manipulatives
While physical manipulatives still have their place in the math classroom, well-designed virtual manipulatives bring an expanded range of capabilities that make math learning more dynamic:
With a connection to the Internet, virtual manipulatives are free of charge and offer anytime access and availability.
Virtual manipulatives are interactive. They can be manipulated and altered in ways that physical objects cannot be. Users can mark, color, highlight, and even reconfigure parts of the virtual object. Users can also input numbers or commands to simulate a sequence of events or create their own mathematical problem. The interactivity of virtual manipulatives lets the user get information and guidance from the text and images on the screen.
Virtual manipulatives link symbolic and iconic notations by saving numerical information or providing mathematical notations that label the on-screen objects.
In addition, the click of a mouse can provide access to unlimited materials, and users simply click on-screen icons for easy clean-up.
Even virtual manipulatives that are based on physical manipulatives have their own unique qualities. For example, a virtual geoboard models the physical geoboard because users can place the bands on the pegs of the board to create geometric shapes. In the virtual environment, however, the bands can be stretched and shaped beyond what is possible in the physical environment, and the areas created by the bands can be colored using a paint palette. These images can then be saved and printed from the computer screen so that the user?s work on the geoboard is not lost.
Some virtual manipulatives allow more open-ended explorations while others provide a guided concept tutorial for users. Those that allow more open-ended explorations can be classified in three groups: pictorial only, simulation, and combined pictorial and numeric.
Pictorial-only applets feature a visual image of the virtual manipulative with no additional information on the screen.
Simulations allow users to run trials that repeat multiple actions on the screen. For example, a student can use a simulation numbers board (or hundreds board) to find the Sieve of Eratosthenes by selecting and running repeated multiples until the numbers board displays all of the prime numbers.
Combined pictorial and numeric applets provide a visual image with accompanying numeric displays that correspond to manipulations of the electronic objects. An example of this is the base-10 blocks applet that displays ones, tens, and hundreds blocks along with a corresponding number sentence for students to solve.
Virtual manipulatives that provide a guided concept tutorial have multiple features for interaction with the user, including:
Directions that verbally guide the user through the steps to solving the exercise.
Numeric information that corresponds to the user?s actions on the objects in the tutorial environment.
Text that provides guiding feedback and correction to help the user compute the on-screen algorithm.
What the Research Is Saying
Research on virtual manipulatives is still in its early stages, but there is promising evidence that the unique properties of virtual manipulatives can have positive effects on student learning.
In a 2005 study, third-graders were placed in physical and virtual manipulative treatment groups to study algebra and rational number concepts. The two groups performed equally well when solving algebraic equations, but there were significant differences in favor of the virtual manipulatives when students learned addition of fractions with unlike denominators. The findings indicated unique characteristics in the design of the virtual fraction applet that influenced student achievement.
A 1996 study of 102 students in grades 2 through 5 demonstrated significant pre- to post-test gains when students used both physical and virtual manipulatives, rather than the physical or the virtual manipulatives alone.
A 2002 study of two sixth-grade classes using virtual and physical geoboards revealed that the physical geoboard might be a more appropriate tool for use in a lesson focused on developing the concept of area, while the virtual geoboard might be a more appropriate tool for a lesson focused on the transformation of shapes to develop one?s own formula for area.
A 2005 study of 19 third-grade students who used several virtual fraction applets in a computer lab during a two-week unit on rational numbers found that virtual manipulatives: 1) helped students learn more about fractions by providing immediate and specific feedback, 2) were easier and faster to use than paper-and-pencil methods, and 3) enhanced students? enjoyment while learning mathematics.
In another 2005 study, kindergarten children used virtual pattern blocks, wooden pattern blocks, and drawings on three different days. When using the virtual pattern blocks, children made a greater number of patterns, used more elements in their pattern stems, and exhibited more creative pattern behaviors.
While these and other research studies confirm the benefits of using virtual manipulatives, researchers caution teachers to use them selectively, with ample doses of teacher direction and guidance. A 2008 study found that number and operations and geometry were the most frequent content areas where virtual manipulatives were used; that virtual geoboards, pattern blocks, tangrams, and base-10 blocks were the virtual manipulatives most commonly used; and that virtual manipulatives were often used by teachers in combination with physical manipulatives.
Guidelines for Selecting Virtual Manipulatives
When selecting a virtual manipulative to use in your mathematics instruction, ask yourself these questions:
What is the mathematics that you wish to teach? Is your mathematical goal conceptual or to develop or practice a skill?
What can the virtual manipulative add to instruction that might not be available or possible in another medium? Can it extend the mathematical possibilities that could be explored beyond what is possible using another tool? Would the use of a combination of virtual and physical manipulatives enhance your students? experiences?
Does the virtual manipulative have the potential to accurately represent the user?s developing understanding of mathematical ideas?
Is the virtual manipulative applet interactive? Are images dynamic and interesting but not too ?busy??
Are the user?s actions accurately reflected in the tool? Can users control the pace of the exploration or task to allow time for mental processing?
Are words, numerals, and pictures used simultaneously to connect representations? Are these representations linked so that the user?s actions on one representation are reflected in other representational forms?
Is constructive feedback provided? Are correct responses confirmed? Are incorrect responses signaled?
Are there hints or suggestions that will help students continue to work on the ideas if they get stuck?
Is the site user friendly, with clear instructions for manipulating dynamic features? Would students be able to engage with the site on their own, or would they need your support?
It may be unlikely that any tool, virtual manipulative or otherwise, will have all of the features suggested above. However, with an understanding of how these tools may support learning to guide your selection, you, as the teacher, can better determine which features are most important for the learning goal and your students.
Where can you go to find high-quality computer-based math manipulatives for grades K?12? Make your first stop the National Library of Virtual Manipulatives (NLVM). With support from the National Science Foundation and based at Utah State University, this decade-long project has developed a free library of interactive virtual manipulatives and concept tutorials for each of the five NCTM content standards.
Click below to browse this resource powerhouse for math teachers and their students!
Since their introduction more than 50 years ago, Unifix Cubes have been used enthusiastically in math classrooms around the world. Next month we celebrate Math Awareness Month by taking a fresh look at this time-honored manipulative and its place in the teaching of foundational math concepts in the early grades.
We'll also be offering a terrific special on a new Unifix grade-level series by math educator Don Balka, not to mention downloadable sample activities from the series. So tune in next month for an issue that's bound to rekindle your love affair with this versatile cube.