Introduction

Born in 1916, Zoltan Dienes is often credited as being the father of mathematical manipulatives. Dienes realized that students often needed to build a concrete understanding of the material they were learning. To do this, he decided to create a physical model that students could manipulate in order to visually see and understand the math that was occurring. Today, manipulatives are defined as any object that students can manipulate in order to learn a concept or mathematical principle. Examples of manipulatives include Dienes blocks, unifix cubes, tangrams, and many more. Since the work of Dienes, much research has been done to show the benefits of building the aforementioned concrete understanding of mathematics, and thus manipulatives have taken a significant place in many mathematics classrooms.

The Dienes blocks, pictured above, which were designed to teach the idea of place value to students, have had enormous success. In fact, many of the students in this class, myself included, probably grew up learning place value with them, before transitioning to the more abstract understanding of the material through the use of numerical symbols. Today, another step is often added in this transition. Students may now be asked to draw the manipulatives on the paper before solving the problem. This step, called the semi-concrete step, helps students equate the numerical symbols with the manipulatives they used. Thus, for the issue of place value, not only learn how to add, but what adding means. In other words, students are not only able to follow the procedures of lining the numbers up by place value and then adding, but they are able to mentally picture what the picture means. Thus, a student with a concrete understanding of the problem is less likely to accept an answer like 4+33 = 433.

However, manipulatives came with their share of problems that will be discussed later but include cost, space and availability. Today, these problems have been greatly reduced thanks to the introduction of virtual manipulatives. A virtual manipulative is a virtual representation of anything that a person can manipulate. For example, a virtual manipulative of the Dienes blocks is shown bellow, taken from http://www.arcytech.org/java/b10blocks/b10blocks.html

Today, virtual manipulatives are powerful tools that may greatly increase student understanding of mathematical concepts. For the first time ever, students are able to create hundreds of examples in merely seconds, while looking at displayed properties, like angle measure, to form and test theories themselves, without simply being asked to memorize existing ones. For example, a student of old was simply told that every triangle has 180 degrees. Today, students may instantly drag points around the screen, creating hundreds of triangles in the process, to see that the angle measure never changes. Students may then conclude on their own that every triangle has 180 degrees. This process helps make math meaningful and memorable for students. Math transforms from boring memorization of abstract principles to a living, breathing, science that is tangible, real, and exciting.

Help with Virtual Manipulatives

The National Library of Virtual Manipulatives has an instruction guide for every manipulative on their site. Simple go to their main site at http://nlvm.usu.edu/ and then you may choose either the specific math content area, the grade level of the manipulatives you are searching for, or both using the table with the word index in the top right cell.

This will generate a list of manipulatives that meet the criteria you selected. For example, If I click the cell in the third row and the forth column, I can see the list of algebra manipulatives for 6-8th graders.

Next, the user may simply select whatever manipulative they are searching for. This will open a Java application of the virtual manipulative with an instructions button on the top right of the screen. Hitting this brings up a how-to guide for the specific manipulative.

Once you figure you how to use the manipulative, create away! I created this triangle as an example.

Other Instructional Site

http://www.mattimath.com/

Role of Virtual Manipulatives in the Classroom

http://www.ct4me.net/math_manipulatives.htm#Role%20of%20Manipulatives

Pros and Cons of Classroom use

As the advantages of physical manipulatives have long been discussed and researched, this will focus on the differences between virtual and physical manipulatives. Virtual manipulatives solve many of the problems often associated with physical

Manipulatives, while creating several new problems. Firstly, virtual manipulatives puts hundreds of manipulatives at your finger tips with simply a few clicks of the mouse. These manipulatives are usually free, and have become easy to find and use. Meanwhile, physical manipulatives often have to be ordered or purchased at teaching specialty stores, and many times, they are overpriced, causing many users to buy the materials to make their own in a time consuming process. On the other side of the argument, however, virtual manipulatives require computers. When being used in the home or with small classes, this is not always a problem, but it may quickly become one in classrooms with many students and limited access to computer labs. Also, using computers help overcome the problem of storing so many manipulatives, but creates the problem of storing so many computers!

Secondly, virtual manipulatives are mess-free. One of the downsides of using physical manipulatives has always been the cleanup afterwards. Students drop pieces on the floor, throw them across the room, and spill big containers of small pieces. With virtual manipulatives, there is no mess or cleanup involved at all.

Thirdly, the virtual manipulatives are safe and prevent classroom disturbances. As students cannot throw or eat virtual manipulatives, using them over physical manipulatives greatly reduces the problems that may arise from these behaviors. On the flip side, however, giving students access to the internet is often dangerous. Students are easily tempted to stray to sites that are unsafe, inappropriate for school, or simply distracting. However, with good internet security blocks and close monitoring, these problems may be overcome.

Next, virtual manipulatives are enabling and limiting at the same time. Using virtual manipulatives opens students and teachers up to activities and uses for the manipulatives they may have never heard of or dreamed of. However, despite the increasingly large numbers of virtual manipulatives out there, the activities one may do with virtual manipulatives are limited to the confines of the program and may stifle creativity. For example, you might have trouble creating three dimensional figures using tangrams, or finding a specific virtual manipulative for your favorite manipulative activity. Also, by having a list of virtual manipulatives at your fingertips, teachers may become content with the activities on a given site, without spending the time to think up new, creative uses for manipulatives.

Finally, using virtual manipulatives allows students to generate numerous examples in short periods of time. It helps guide students through math, creates “picture proofs,” and allows students to manipulate math in a way that had been impossible prior to the release of virtual manipulatives in the 1990’s.

Overall, research has shown that virtual manipulatives may increase the creativity, complexity and number of solutions attempted by the student. Virtual manipulatives may help guide students to use the desired strategies in solving the problem, facilitating the development of deeper understanding of the mathematics behind the problems.

Dissertation Research on Virtual Manipulatives

http://64.233.169.104/search?q=cache:oHF9hv14ursJ:ncsmonline.org/docs/StLouisSessionHandouts/Virtual_Manip_Res_PP_2006.pdf+teachers+using+virtual+manipulatives&hl=en&ct=clnk&cd=19&gl=us&client=firefox-a

Resource Review of NCTM

http://homeschooling.families.com/blog/math-resource-review-national-library-of-virtual-manipulatives-and-alekscom

Classroom Examples

Due to the facts that virtual manipulatives are still relatively new and that their design is such that teachers just go to the sites and use them without creating lesson plan sites of creative implementation, locating many such sites proves to be a challenge. However, here are a few notable ones.

This is a Virginia designed lesson plan aimed at sixth grade students. The lesson plan uses the NLVMIM website to guide students to an understanding of histograms. This lesson plan is useful because it not only shows one method of teaching SOL material in meaningful ways, but it also shows the procedural requirements for designing and implementing a virtual manipulative lesson.

http://www.knowledge.state.va.us/cgi-bin/lesview.cgi?idl=1033

This site provides descriptions of lesson plan instructions, methods, objectives, activities and assessment for use with virtual manipulatives. This site is a great example because it shows many different components that go into any good lesson design and it shows how the fit in with a virtual lesson plan design.

http://www.mattimath.com/

This site is the Boston Public School systems response to the No Child Left Behind act. It is designed to enhance the curriculum with technology and shows how to seamlessly integrate the book and curriculum with virtual manipulatives and technological tools. This site is a great example of how to balance a lesson with all of the necessary components and resources without getting overly carried away with a single component.

http://boston.k12.ma.us/teach/technology/select/

This grade six site contains several useful manipulatives. More importantly, however, at the top of every manipulative the site seeks to answer two important questions, “What you will do” and “How will you use it?” This is a great example of how the use of virtual manipulatives must be purposeful and meaningful. If these two questions cannot be answered, then perhaps the virtual manipulative should not be used.

http://harcourtschool.com/activity/elab2002/grade_6/index_ca.html

NCTM’s Views

http://my.nctm.org/eresources/article_summary.asp?URI=TCM2002-02-372a

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This work is licensed by Sam Rhodes under a Creative Commons Attribution License (CC-BY 2.0), and is an Open Educational Resource.

Last edited by Sam Rhodes on Oct 20, 2008 7:39 am GMT-5.