1. I. Introduction A) What are Manipulatives? B) What are Virtual Manipulatives (VMs)? II. The Way Things Are A) Perceived Benefits of VMs B) Collections of VMs C) Research on VMs D) Suggestions for development, use and research III. What’s Missing A) Virtual Schools B) Secondary Ed C) Further Research Outline

2. I.A - What are Manipulatives? Physical objects used to promote learning in math education Well-established as being valuable educational tools for mathematics 1 Do not guarantee success 2 Can be used for rote learning 3 1. (Sowell 1989) 2. (Clements & McMillen 1996; Reimer & Moyer 2005) 3. (Clements & McMillen 1996; Reimer & Moyer 2005; Durmus & Karakirik 2006)

3. "... an interactive, Web-based visual representation of a dynamic object that presents opportunities for constructing mathematical knowledge." (Moyer, Bolyard, & Spikell 2002, p.373)  Graphs of Linear Equations Graphs of Linear Equations  Distance Vs. Time Graphs Distance Vs. Time Graphs I.B - What are Virtual Manipulatives?

4.  Easily accessible and manageable 1  Low technological learning curve 2  Available out-of-school 3  Many people can develop them 4  Can do things that are not possible with Physical 5 Manipulatives, pencil and paper, or other tools Logistic Benefits: 1. (Clements & McMillen 1996, p.27; Dorward 2002, pg. 330; Heath 2002, p. 43; Leathrum 2001; Moyer & Bolyard 2002, p.20; Moyer, Bolyard, & Spikell 2002, p.375) 2. (Gadanidis, Gadanidis, & Schindler 2003, p. 323; Heath 2002, p. 43; Roschelle, DiGiano, & Chung 2000) 3. (Dorward 2002, pg. 330; Heath 2002, p. 44; Keller, Wasburn-Moses & Hart 2002; Moyer & Bolyard 2002 p.20; Moyer, Bolyard, & Spikell 2002, p.375) 4. (Leathrum 2001) 5. (Clements & McMillen 1996, p.274; Crawford & Brown 2003, pg. 172; Forster 2006; Keller, Wasburn-Moses & Hart 2002; Reimer & Moyer 2005, p.5-6) II.A - Perceived Benefits of VMs

5.  Instantaneous, corrective feedback 1  Various representations of concepts 2  Link symbolic concepts with concrete representations 3  Link specific ideas to general understandings 4  Promotes problem solving and the ability to test hypotheses 5  Helpful for students with disabilities 6  Increases attention and motivation (even in teachers) 7 Teaching Benefits: 1. (Clements & McMillen 1996, p.271; Crawford & Brown 2003, pg. 172; Durmus & Karakirik 2006; Reimer & Moyer 2005, p.5-6; Suh & Moyer 2005, p. 5) 2. (Clements & McMillen 1996, p.271, 274; Heath 2002, p. 44; Keller, Wasburn-Moses & Hart 2002; Moyer & Bolyard 2002, p.20; Reimer & Moyer 2005, p.5-6; Suh & Moyer 2005, p. 5) 3. (Clements & McMillen 1996, p.273; Reimer & Moyer 2005, p.5-6; Suh & Moyer 2005, p. 5) 4. (Clements & McMillen 1996, p.270; Durmus & Karakirik 2006; Jacobs 2005; Moyer & Bolyard 2002, p.20; Suh & Moyer 2005, p. 5) 5. (Clements & McMillen 1996, p.276; Durmus & Karakirik 2006; Gadanidis, Kamran, & Hai-Ning 2004, p.277; Heath 2002, p. 44; Jacobs 2005; Suh & Moyer 2005, p. 5) 6. (Miller, Brown, & Robinson 2002; Riley, Beard, & Strain 2004) 7. (Clements & McMillen 1996, p.276; Leathrum 2001; Reimer & Moyer 2005, p.5-6) II.A - Perceived Benefits of VMs

6.  Mathlets (JOMA) Mathlets  Illuminations (NCTM) Illuminations  ESCOT ESCOT  NLVM NLVM  Project Interactivate Project Interactivate 2) Formal Project Collections: 3) For Profit Collections ExploreLearning 1) Individual Developer Collections II.B - Existing Collections of VMs

7. Problems with current collections: Redundancy 1 Not Designed for Reuse 2 Teacher’s lack understanding of how to find and use them 3 1. (Roby 2001) 2. (Roschelle, DiGiano, & Chung 2000 p.666) 3. (Suh & Moyer 2005) II.B - Existing Collections of VMs

8.  Three were classroom studies,  Two found some benefits 1, One found no difference 2  Three were studies of teacher use and attitudes 3  mixed feelings and little experience  One purported to show that Java-Applets could do things that graphing calculators could not 4 Of the seven articles describing actual research projects: 1. (Reimer & Moyer, 2005; Suh & Moyer 2005) 2. (Dorward 2002) 3. (Keller, Wasburn-Moses, & Hart 2002; Gadanidis, Gadanidis, & Schindler 2003; Crawford & Brown 2003) 4. (Forster 2006) II.C - Existing Research on VMs

9.  Cholmsky (2003) and Reimer & Moyer (2005) used Marzano’s (1998) meta-analysis of instructional methods that work  Clements and McMillen (1996) used Piaget and Holt to argue that computer manipulatives are no less concrete than physical. Two alternative arguments to support use of VM’s with research II.C - Existing Research on VMs

10.  "Although these results are mixed, the amount of research on high-quality dynamic virtual manipulatives is so limited that a judgment about their potential uses in mathematics instruction is entirely speculative". (Reimer & Moyer 2005, p. 8) Conclusion: II.C - Existing Research on VMs

11.  Based on sound Pedagogy 1  Influenced by HCI research 1  Context embedded 1  Classroom testing and feedback 2  Reusable 3  Set of Standards 4 1. (Gadanidis, Kamran, & Hai-Ning 2004) 2. (Gadanidis, Kamran, & Hai-Ning 2004; Keller, Wasburn-Moses & Hart 2002; Moyer, Bolyard, & Spikell 2002) 3. (Roschelle, DiGiano, & Chung 2000; Roschelle, Hand, DiGiano 2001) 4. (Roby 2001) II.D - Suggestions for Development of VMs

12.  Teachers need to be involved and direct use 1  Students should have opportunity to reflect and explore 2  Teachers need more training 3  Teachers need to share results with each other and developers 4 1. (Reimer & Moyer, 2005; Suh & Moyer 2005) 2. (Clements & McMillen 1996) 3. (Gadanidis, Gadanidis, Schindler 2003; Suh & Moyer 2005) 4. (Leathrum 2001; Roby 2001; Moyer, Bolyard, & Spikell 2002) II.D - Suggestions for Use of VMs

13.  Incorporating HCI theory in design 1  Influence of context 2  Teacher awareness and training 3 1. (Gadanidis, Kamran, & Hai-Ning 2004) 2. (Gadanidis, Kamran, & Hai-Ning 2004) 3. (Crawford & Brown 2003; Gadanidis, Gadanidis, Schindler 2003; Reimer & Moyer 2005) II.D - Suggestions for Research on VMs

14.  Virtual Schools  The online nature of VMs make them suitable for virtual schools  Face-to-face and online teaching methods differ  Virtual Schools could offer studies in which more students could be used. III.A - What’s Missing?

15.  Secondary Education  Most existing applets are focus on elementary, middle-school and undergraduate topics.  Virtually all of the existing research deals with elementary and middle school age students and topics. III.B - What’s Missing?

16.  Future Research:  What aspects of some VMs make them more effective than others?  Is there a point in a lesson where they are most useful?  How can teachers be better educated on how and when to use VMs?  What topics may be most beneficial to be taught with VMs?  What are the effects of using VMs Vs. physical manipulatives (especially for very young students)? III.B - What’s Missing?

17. Although very little is conclusively known about them, Virtual Manipulatives are a very promising tool for mathematics education. As use, development, and research on VMs increases they will become more effective teaching tools. THE END (References on next page)

18. Cholmsky, P. (2003). Why Gizmos Work: Empirical Evidence for the Instructional Effectiveness of ExploreLearning's Interactive Content. Retrieved April 8, 2006 from: http://www.explorelearning.com/View/downloads/WhyGizmosWork.pdf Clements, D.H. & McMillen, S. (1996). Rethinking "Concrete" Manipulatives. Teaching Children Mathematics. 2(5), 270-279. Crawford, C. & Brown, E. (2003). Integrating Internet-based Mathematical Manipulatives Within a Learning Environment. Journal of Computers in Mathematics and Science Teaching. 22(2), 169-180. Dorward, J. (2002). Intuition and Research: Are They Compatible? Teaching Children Mathematics. 8(6), 329- 332. Durmus, S. & Karakirik, E. (2006). Virtual Manipulatives in Mathematics Education: A Theoretical Framework. The Turkish Journal of Educational Technology. 5(1). Retrieved April 8, 2006 from: http://www.tojet.net/articles/5112.htm Forster, P. A. (2006). Assessing technology-based approaches for teaching and learning mathematics. International Journal of Mathematical Education in Science and Technology. 37(2), 145-164. Gadanidis, G., Gadanidis, J., & Schindler, K. (2003). Factors Mediating the Use of Online Applets in the Lesson Planning of Preservice Mathematics Teachers. The Journal of Computers in Mathematics and Science Teaching, 22(4), 323-344. Gadanidis, G., Kamran, S., & Hai-Ning, L. (2004). Designing Online Mathematics Investigation. The Journal of Computers in Mathematics and Science Teaching, 23(3), 275-98. Heath, G. D., (2002) Using Applets in Teaching Mathematics. Mathematics and Computer Education. 36(1), 43- 52. References 1 of 3

19. Hodgson, T. (2006). Applets and Algebra in the Middle Grades. ON-Math. 4(1). Retrieved April 8, 2006 from: http://my.nctm.org/eresources/view_article.asp?article_id=7437&page=1 Illuminations Web site. (2006). Retrieved April 8, 2006 from: http://illuminations.nctm.org/ Jacobs, K.L. (2005). Investigation of Interactive Online Visual Tools for the Learning of Mathematics. International Journal of Mathematical Education in Science and Technology. 36(7), 761-768. Keller, B., Wasburn-Moses, J., & Hart, E. (2002). Improving Student's Spatial Visualization Skills and Teacher's Pedagogical Content Knowledge by using On-Line Curriculum-Embedded Applets. National Council of Teachers of Mathematics Illuminations Project. Retrieved April 8, 2006 from: http://illuminations.nctm.org/downloads/IsoPaperV4.pdf Leathrum, T. (2001). Writing Mathlets I: A Call to Math Professionals. Journal of Online Mathematics and its Applications. 1(2). Retrieved April 8, 2006 from: http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=460 Marzano, R.J. (1998). A theory-based meta-analysis of research on instruction. Aurora, CO: Mid- continent Research for Education and Learning. Marzano, R.J., Gaddy, B.B., & Dean, C. (2000). What works in classroom instruction. Aurora, CO: Mid-continent Research for Education and Learning. Miller, D., Brown, A., Robinson, L. (2002). Widgets on the Web, Using Computer-Based Learning Tools. Teaching Exceptional Children. 35(2), 24-28. Moyer, P.S. & Bolyard, J.J. (2002). Exploring Representation in the Middle Grades. Australian Mathematics Teacher. 58(1), 19-25. References 2 of 3

20. References 3 of 3 Moyer, P.S., Bolyard, J.J. & Spikell, M.A. (2002). What Are Virtual Manipulatives? Teaching Children Mathematics. 8(6), 372-377. National Library of Virtual Manipulatives. (2006). Retrieved April 8, 2006 from: http://enlvm.usu.edu/ma/nav/doc/intro.jsp. On-Math Web site. (2006). Retrieved April 8, 2006 from: http://my.nctm.org/eresources/journal_intro.asp?journal_id=6 Project Interactivate web site. (2006). Retrieved April 8, 2006 from: http://www.shodor.org/interactivate/. 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. Riley, G., Beard, L.A., & Strain, J. (2004). Assisstive Technology at Use in the Teacher Education Programs at Jacksonville State University. Tech Trends. 48(6) 47-49. Roby, T. (2001). The JOMA Mathlet Project. Journal of Online Mathematics and its Applications. 1(1). Retrieved April 8, 2006 from: http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=352 Roschelle, J., DiGiano, C., & Chung, M. (2000). Reusability and Interoperability of Tools for MAthematics Learning: Lessons from the ESCOT Project. Proceedings of Intelligent Systems & Applications at University of Wollongong, NSW Australia. ICSC Academic Press, Wetaskiwin, AB, pp. 664-669. Roschelle, J. Hand, V. & DiGiano, C. (2001). Towards a Coherent On-Line Collection of Tools. Journal of Online Mathematics and its Applications. 1(2). Retrieved April 8, 2006 from: http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=461 Suh, J. & Moyer, P. S. (2005). Examining Technology Uses in the Classroom: Developing Fraction Sense Using Virtual Manipulative Concept Tutorials. Journal of Interactive Online Learning. 3(4).