WHAT ARE VIRTUAL MANIPULATIVES?

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1 by SCOTT PIERSON AA, Community College of the Air Force, 1992 BS, Eastern Connecticut State University, 2010 A VIRTUAL MANIPULATIVES PROJECT SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR TECHNOLOGY APPLICATIONS IN MATHEMATICS (EDU 556) Instructor: Dr. Hari P. Koirala DEPARTMENT OF EDUCATION EASTERN CONNECTICUT STATE UNIVERSITY July, 2014

2 Introduction Virtual manipulatives have been broken down into two separate types (Moyer, Bolyard, & Spikell, 2002): static and dynamic. Static manipulatives are visual images ordinarily associated with pictures in books, drawings on an overhead projector, sketches on a chalkboard, etc. These concrete depictions of a problem are in no way interactive, they can only be viewed as presented. Students can not turn, flip, or otherwise maneuver them to see them from a different perspective. Dynamic manipulatives, however, are concrete objects that can be maneuvered, but instead of maneuvering them by hand as you would a literal object, these items are moved with a computer mouse. For the purpose of this study, this latter type will be what is referred to when we use the term virtual manipulative. Given the information above and for the purpose of this paper, we will define a virtual manipulative as an interactive, Web based visual representation of a dynamic object that presents opportunities for constructing mathematical knowledge (Moyer, Bolyard, & Spikell, 2002, p. 373). In my limited teaching experience I have never used a virtual manipulative for the purpose of educating a student and at no time has anyone used a virtual manipulative to assist in my own education. Therefore, this is a new area for me to explore! I can say that from my previews of a couple of different manipulative websites, this technology offers a fun and interesting supplement to a teachers curriculum. While I don t have a class or students, I can easily see how virtual manipulatives and online software (VM/OS) can play a role in the teaching of mathematics to students of all levels. Not only do the games and activities make the lessons fun, but the constant

3 repetition provided through these manipulatives will only continually reinforce the material. Another benefit I noted was that immediate positive of negative feedback is provided in many instances, allowing a student to learn by trial and error, but privately in order to avoid making a mistake in front of peers. To add additional support for the use of VM/OS, the National Council of Teachers of Mathematics (NCTM) emphasizes the importance of VM/OS. At the NCTM website 21 E examples are linked to NCTM standards for grades Pre K 12. Each activity includes an explanation of the mathematics standard, instructions for using the manipulative or interactive object, a problem based lesson and reflective activities to extend the lesson. The NCTM site also contains the Illuminations page, including more than 100 manipulatives, lessons based on NCTM standards and links to related online activities. The purpose of Illuminations is to provide Standards based resources that improve the teaching and learning of mathematics for all students and provide materials that illuminate the vision for school mathematics set forth in Principles and Standards for School Mathematics. Oddly, in my search for any reference to Web based manipulatives in the Common Core State Standards (CCSS), I found no mention (CCSS Staff, n.d.). However, several organizations have developed or promote the use of various manipulatives. I found several very helpful virtual manipulatives developed by Utah State University for their National Library of Virtual Manipulatives. This site is well organized and lays out dozens of excellent manipulatives, organized by grade and topic. One such tool (seen to the right) helps first grade students tell and write time in hours and half hours using analog and digital clocks. (CCSS Staff, n.d., 1.MD.B.3).

4 There are several reasons to use manipulatives. One aspect is the time, cost and space savings over use of concrete manipulatives. Virtual manipulatives are free, available to the entire class, and working with them does not require tables or other large spaces. Research (Moyer Peckenham, Salkind, & Bolyard, 2008) has shown that concrete manipulatives help students better understand abstract concepts in math, and comparative studies show virtual manipulatives to be as effective as concrete ones (Donovan, 2008). In my research for this topic, I found several articles that I found to be particularly helpful. The first and most notable was a research project regarding virtual manipulatives used by K 8 teachers (Moyer Peckenham, Salkind, & Bolyard, 2008). While I initially expected this study to focus on the success of virtual manipulatives, rather it discussed teacher strategies and trends in using the manipulatives. Researchers analyzed 95 lesson summaries in which classroom teachers described their uses of virtual manipulatives during school mathematics instruction. I found it most interesting that the most commonly found use was to use the virtual manipulative alone or to use the physical manipulative first, followed by the virtual manipulatives. Another important finding of this study was that teachers used the virtual manipulatives during the main portion of their lessons when students were learning mathematics content. As the newly appointed Technology Director of the school on whose board I serve, I may find myself in a situation where the use of virtual manipulatives is needed. Knowing how and when to use the manipulative may be as important as which one to use. Another very helpful article (Donovan, 2008) discussed the use of virtual manipulatives in math instruction. Beginning with the same referenced definition of a virtual manipulative that I had found above, this article discussed many helpful topics such

5 as factors to consider when selecting a virtual manipulative and it even listed several manipulatives, to include sample problems from the sites. Project The project I have selected will demonstrate how one of the many manipulatives in Utah State University s National Library of Virtual Manipulatives can help students understand numbers and operations. More specifically, this project has the following objectives: 1. Encourage logical analysis 2. Reinforce one place decimal addition skills Overview of the VM/OS The project is a one place decimal addition puzzle called Circle 3. This project has been selected for a 5th intended for all Grade math class and is students in the class. The Circle 3 puzzle will encourage students to analyze the puzzle logically in order to fit the puzzle so that the complete circle add up the nine numbers correctly into three numbers within each to three. The project will also help reinforce one place decimal addition skills and will meet the NCTM Standard Criteria for grades 3 5 by Communicat[ing] about math using games, use simulation software to investigate distance, rate, and time; investigate data. (NCTM Staff, n.d., 5.1) Project Activities Each student will be provided a laminated sheet containing one large set of circles as pictured above, except the numbers will have been removed. Students will also be

6 provided a blue and black erasable sharpie so that they can follow along with the problems on their own sheet. For students requiring differentiation, the circles may be partially completed as needed. Students may also be assigned to work one on one with a teacher first, during, or after the class exercise to ensure they receive the instruction necessary to complete the task. On the reverse of the laminated sheet will be a simple rubric so that all students know what is expected of them. The blue numbers are already in position when the manipulative loads and cannot be moved. The remaining numbers (from 0.4 through 1.7) can be dragged into (or removed from) any empty space in the ring of circles. When the sum of the numbers in a circle is exactly 3.0, then the whole circle changes color. But it isn't enough to make a few circles in the ring add to 3. There may be several ways to make the first couple of circles sum to 3, but there is only one way to combine all the given numbers so that every circle sums to 3. One can drag numbers into a couple of target circles, but if a given choice appears to lead to a dead end, you can "back up" by removing any of the black numbers you've placed. It is a good idea to play freely at first, but pencil and paper are helpful to analyze what is needed to color the whole ring of circles. The New Game button provides another "seeding" of five blue numbers, so that a great deal of practice is available by repeated play. As students gain confidence and skill, it would be possible for two or more to race against each other or to record times but that would be for a different project.

7 Solve the puzzle by dragging the black numbers to the blank spaces. Remember, you cannot move the blue numbers. Four puzzles will be completed. The first puzzle will be completed by the entire class as one large group. The second and third puzzles will be completed by small groups. The fourth puzzle will be completed individually. Project Assessment The first assessment will be when the circles change (or don t change) color. The second assessment will be based on a possible 100 points for each student, where 25 points will be awarded for each successfully completed puzzle. Partial credit will be awarded for circles that contain numbers equaling 3, even if the puzzle is not completed successfully.

8 Feedback During the project, the teacher will take note of areas that where students seemed to struggle or had a hard time understanding. Following the project, students will be asked to provide feedback on the aspects of the project they found most difficult. Students will also be asked how the directions could have been made easier to understand. Final Thoughts I have never taught this project before but I have practiced it three times and found it to be fun and an interesting way to learn how to analyze a puzzle and consider all the options. My one concern is for students who struggle with the project. Since I have never taught a learning disabled child, I would need to seek help from someone more familiar with assistive teaching methods.

9 REFERENCES CCSS Staff. (n.d.). Common Core State Standards for Mathematics. 1.MD.B.3. Retrieved from CoreStandards.Org: CCSS Staff. (n.d.). Common Core State Standards in Mathematics. Retrieved from CoreStandards.Org. Donovan, J. (2008, October 1). Teaching Mathematics with Virtual Manipulatives. Retrieved from Educators Ezine: theclassroom/0015/teaching mathematics with virtual manipulatives/45397 Moyer, P., Bolyard, J., & Spikell, M. (2002). What Are Virtual Manipulatives. Teaching Children Mathematics, Retrieved from: ecsu.ct.edu/bbcswebdav/pid dt content rid _1/courses/ECSU /Moyeretal 2002.pdf Moyer Peckenham, P., Salkind, G., & Bolyard, J. (2008). Virtual Manipulatives Used by K 8 Teachers for Mathematics Instruction: Considering Mathematical, Cognitive, and Pedagogical Fidelity. Contemporary Issues in Technology and Teacher Education, 8(3), Retrieved from NCTM Staff. (n.d.). E Examples from Principles and Standards for School Mathematics. Retrieved from National Council of Teachers of Mathematics:

10 WORKSHEET RUBRIC SCORE Mathematical Knowledge (Do I know it?) Strategic Knowledge (How do you plan?) Explanation (Can you explain it?) 4 All my circles added up to 3. I used math words correctly to show understanding. I did the work with no mistakes. I figured out how to solve the problem and make the circles add up to 3. I showed the steps I used to get the right answers. I wrote or explained what I did and why I did it. I explained my completed problem. 3 All my circles added up to 3. I used most of the math words correctly. I did the work with only a few mistakes, which I fixed. I figured out the problem with a little guessing. I can show most of the steps I used to get the right answer. I wrote or explained mostly about what I did. I wrote a little about why I did it. I explained most of the process. 2 I understand how to do the work but not all of my circles added up to 3. I used some of the math words correctly. I had trouble completing the assignment without making a lot of mistakes. I figured out some of the problem. I showed some of the steps but the plan wasn t clear. I wrote or explained a little about what I did or why, but not both. I can explain a little about the process. 1 I tried to do the problem but I really didn t understand it. I couldn t figure out how to get the circles to add up. I couldn t show the steps I took. I didn t write or explain how I got the answers I got. I really didn t understand the process. 0 I didn t try to answer the problem. I didn t show a plan. I didn t write or explain anything.

11 CIRCLE 3 Navigate to: You will be presented with a puzzle similar to this: Your task is to use the 9 one-decimal numbers to the right of the puzzle to make each of the 7 complete circles equal 3.

12 You will notice in the sample above that 0.8 and 1.2 have been added to the circle containing the blue number 1.0. (Note that blue numbers cannot be moved.) Additionally, note that 1.3 has been added to the circle that contains the black 1.2 we moved above and the blue 0.5 that was already in the circle. Also note how successfully completed circles (where the numbers add up to 3) change color.

13 Note in the item below that 1.4 and 1.1 have been added to the other circle that contained 0.5.

14 Note in the sample above that each of the seven circles contains numbers that add up to 3. Note that successful completion of the puzzle changes the color to red. NOTES ABOUT THIS PROJECT: You will complete four puzzles. The first puzzle you will complete as a group while the teacher demonstrates how to move the numbers in and out of the circles. The second and third puzzles will be completed in your pre-assigned small groups. The last puzzle will be completed individually. Each successfully completed circle will receive 25 points. Partial credit will be given for circles that contain numbers adding up to 3.0, even if the puzzle is not complete. Please ensure the teacher sees your completed circle before selecting New Game. There will be plenty of time to successfully complete the puzzles so do not make mistakes because you feel you need to hurry.