1. Virtual Manipulatives & Online Collaboration
with the free math software GeoGebra
Markus Hohenwarter Florida Atlantic University
Markus Hohenwarter,

2. NSF MSP FAU-Broward
NSF Math & Science Partnership Florida Atlantic University & Broward County Schools ( ) PIs: Heinz-Otto Peitgen, Richard Voss
Master’s program for Middle School Mathematics Teachers at FAU
Focus on Math Content Knowledge & Technology Integration into Teaching
Markus Hohenwarter,

3. Outline Virtual Manipulatives Online Collaboration with GeoGebra
What are Virtual Manipulatives?
What we Know
What‘s Missing
Online Collaboration with GeoGebra
GeoGebra – free dynamic math software
GeoGebraWiki – reusable & changeable VMs
International GeoGebra Institute
Markus Hohenwarter,

4. What are Virtual Manipulatives?
Markus Hohenwarter,

5. What are Manipulatives?
Physical objects used to promote learning e.g. tangram, base ten blocks
Well-established as being valuable education tools for mathematics (Sowell 1989)
Markus Hohenwarter,

6. What are Virtual Manipulatives?
“… an interactive, web-based visual representation of a dynamic object that presents opportunities for constructing mathematical knowledge”
(Moyer, Bolyard, & Spikell 2002)
Web-based
Manipulable
Synonyms: Math Applets, Mathlets, Widgets, Gizmos
Markus Hohenwarter,

7. Student Activity
“Mathematics is not, as has been said, a spectator sport. Too much of current instruction fails to actively involve students.”
(National Library of Virtual Manipulatives)
Markus Hohenwarter,

8. Example: Multiplying Fractions
Animation (Cyberlearning.org)
Web-based
NOT manipulable
Virtual Manipulative (GeoGebra)
Manipulable
Markus Hohenwarter,

9. Examples: Virtual Manipulatives
Slope-Intercept Form Angles in a Clock
Markus Hohenwarter,

10. Collections of K-12 VMs National Library of Virtual Manipulatives
NCTM Illuminations
MERLOT Mathematics Portal
Project Interactive
MathTools
GeoGebraWiki
Markus Hohenwarter,

11. Virtual Learning What we Know
Markus Hohenwarter,

12. Which medium is best?
“What we have learned from all the media comparison research is that it’s not the medium, but rather the instructional methods that cause learning.
When the instructional methods remain essentially the same, so does the learning, no matter how the instruction is delivered.”
(Clark & Mayer 2007: e-Learning and the Science of Instruction)
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13. Web-Based Virtual Learning
“Student achievement in courses using and web-based virtual learning is slightly better than in courses using traditional instruction.”
(Meriti 2006: Technology in Schools: What research says)
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14. (Meriti 2006: Technology in Schools: What research says)
“Overall, across all uses in all content areas technology does provide a small, but significant, increase in learning when implemented with fidelity.”
(Meriti 2006: Technology in Schools: What research says)
Markus Hohenwarter,

15. Virtual Manipulatives What we Know
Markus Hohenwarter,

16. Logistic Benefits of VMs
Easily accessible and manageable
Low technological learning curve
Available out-of-school
Many people can develop them
Can do things that are not possible with physical manipulatives or pencil and paper
(Young 2006)
Markus Hohenwarter,

17. Example: Reflection at Line
Markus Hohenwarter,

18. Teaching Benefits of VMs
Instantaneous, corrective feedback
Various representations at the same time
Help students to understand abstract concepts
Lead to more complex, richer understanding
Promote problem solving and the ability to test hypotheses
Helpful for students with disabilities
Increase attention and motivation (also in teachers)
(Young 2006, CITEd 2007)
Markus Hohenwarter,

19. Teachers are important
“The mere use of manipulatives does not guarantee that students understand concepts and procedures and be able to connect these concepts to abstract symbols without teachers making these connections explicit.”
(Reimer & Moyer 2005)
Markus Hohenwarter,

20. Virtual Manipulatives What‘s Missing?
Markus Hohenwarter,

21. What’s missing for VM?
Professional Development: Support for teachers on how to find and use them
Reusable and Changeable VMs: Teachers should be able to adapt VMs to their specific needs
Research: More classroom studies, especially in high schools
Markus Hohenwarter,

22. Outline Virtual Manipulatives Online Collaboration
What are Virtual Manipulatives?
What we Know
What‘s Missing
Online Collaboration
GeoGebra – free dynamic math software
GeoGebraWiki – collaborative VMs
International GeoGebra Institute
Markus Hohenwarter,

23. Online Collaboration with the free math software GeoGebra
Markus Hohenwarter,

24. What is GeoGebra? Dynamic Mathematics Software
For Learning and Teaching in Schools
Geometry, Algebra and Calculus
Open Source
Markus Hohenwarter,

25. GeoGebra = Geometry + Algebra Algebra Window
Geometry & Graphics Window
Markus Hohenwarter,

26. Educational Principles
Student Activity
Multiple Representations
Experimental and Guided Discovery Learning
Markus Hohenwarter,

27. Markus Hohenwarter, FAUMarkus. Hohenwarter@sbg. ac. at, 2004 - www
Markus Hohenwarter,

28. Markus Hohenwarter, FAUMarkus. Hohenwarter@sbg. ac. at, 2004 - www
Markus Hohenwarter,

29. Markus Hohenwarter, FAUMarkus. Hohenwarter@sbg. ac. at, 2004 - www
Markus Hohenwarter,

30. GeoGebra Virtual Manipulatives
Every GeoGebra figure can be exported as a web-page (html)
So called Dynamic Worksheet
Markus Hohenwarter,

31. Open Source & Online Collaboration
Markus Hohenwarter,

32. Why Open Source? Available free of charge in Schools and at Home
Online Collaboration
International Community
User Forum – Users help Users
GeoGebraWiki – Pool of Shared Materials
Expandable by using other Open Source Libraries
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33. World Wide User Community
35 languages
190 countries
300,000 visitors per month
Markus Hohenwarter,

34. GeoGebra User Forum www.geogebra.org/forum
Markus Hohenwarter,

35. GeoGebraWiki – Free Materials
Markus Hohenwarter,

36. International GeoGebra Institute (IGI)
IGI Activities
Offers free GeoGebra training for teachers
Issues GeoGebra certification
Nurtures a community of researchers
IGI supports you to
Establish an IGI site at your university
Join projects and collaborate with researchers
Write grant proposals
Participate in IGI conferences
Markus Hohenwarter,

37. What’s missing for VMs? Reusable and Changeable Virtual Manipulatives
→ Dynamic GeoGebra Worksheets, GeoGebraWiki
Professional Development
→ MSP projects, IGI workshops
Research
→ MSP collaboration, IGI
Markus Hohenwarter,

38. References 1/2
CITEd Research Center (2007). Learning Mathematics with Virtual Manipulatives. Retrieved Dec 2007 from
Clark, R. C., & Mayer, R. E. (2007).  E-learning and the science of instruction (2nd ed). San Francisco: Jossey-Bass. 
Hohenwarter, M., & Preiner, J. (2007). Dynamic mathematics with GeoGebra. Journal of Online Mathematics and its Applications. ID 1448, vol. 7, March 2007
Hohenwarter, M., & Preiner, J. (2007). Creating mathlets with open source tools. Journal of Online Mathematics and its Applications. ID 1574, vol. 7, August 2007
Hohenwarter, M., & Lavicza, Z. (2007). Mathematics teacher development with ICT: towards an International GeoGebra Institute. Proceedings of the British Society for Research into Learning Mathematics. 27(3): University of Northampton, UK
Markus Hohenwarter,

39. References 2/2
Metiri Group (2006): Technology in Schools: What research says. Retrieved Dec 2007 from
Moyer, P.S., Bolyard, J.J. & Spikell, M.A. (2002). What Are Virtual Manipulatives? Teaching Children Mathematics. 8(6),
Reimer, K., & Moyer, P.S. (2005). Third-Graders Learn About Fractions Using Virtual Manipulatives: A Classroom Study. Journal of Computers in Mathematics and Science Teaching. 42(1), 5-25.
Sowell, E. J. (1989). Effects of Manipulative Materials in Mathematics Education. Journal for Research in Mathematics Education. 20(5) pp
Young, D. (2006). Virtual Manipulatives in Math Education. Retrieved Dec 2007 from
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40. The instructional practices and assessments discussed or shown in these
presentations are not intended as an endorsement by the U.S. Department of Education.