International Perspectives on Teaching and Learning Mathematics with Virtual Manipulatives pp 3-23 | Cite as

# Revisiting the Definition of a Virtual Manipulative

## Abstract

In 2002, Moyer, Bolyard and Spikell defined a virtual manipulative as an “an interactive, Web-based visual representation of a dynamic object that presents opportunities for constructing mathematical knowledge” (p. 373). The purpose of this chapter is to revisit, clarify and update the definition of a virtual manipulative. After clarifying what a virtual manipulative is and what it is not, we propose an updated definition for virtual manipulative: *an interactive, technology*-*enabled visual representation of a dynamic mathematical object, including all of the programmable features that allow it to be manipulated, that presents opportunities for constructing mathematical knowledge*. The chapter describes the characteristics of five of the most common virtual manipulative environments in use in education: single-representation, multi-representation, tutorial, gaming and simulation.

## Keywords

Mathematical Knowledge Number Line Mathematical Object Programmable Feature Multiple Representation## References

- Anderson-Pence, K. L. (2014).
*Examining the impact of different virtual manipulative types on the nature of students’ small-group discussions: An exploratory mixed-methods case study of techno-mathematical discourse*(Doctoral dissertation). ProQuest Dissertations and Theses database (UMI No. 3683422).Google Scholar - Ares, N., Stroup, W. M., & Schademan, A. R. (2008). The power of mediating artifacts in group-level development of mathematical discourses.
*Cognition and Instruction,**27*(1), 1–24. doi: 10.1080/07370000802584497 CrossRefGoogle Scholar - Barendregt, W., Lindström, B., Rietz-Leppänen, E., Holgersson, I., & Ottosson, T. (2012). Development and evaluation of Fingu: A mathematics iPad game using multi-touch interaction. In H. Schelhowe (Ed.),
*Proceedings of the 11th international conference on interaction design and children*(pp. 204–207). ACM, New York, NY. http://doi.org/10.1145/2307096.2307126 - Bolyard, J. J., & Moyer, P. S. (2007, March).
*Selecting dynamic technology representations for mathematics teaching*. Research presentation, 85th annual meeting of the national council of teachers of mathematics (NCTM), Atlanta, GA.Google Scholar - Bos, B. (2009a). Technology with cognitive and mathematical fidelity: What it means for the Math classroom.
*Computers in the Schools,**26*(2), 107–114. doi: 10.1080/07380560902906088 CrossRefGoogle Scholar - Bos, B. (2009b). Virtual math objects with pedagogical, mathematical, and cognitive fidelity.
*Computers in Human Behavior,**25*, 521–528.CrossRefGoogle Scholar - Carr, J. M. (2012). Does math achievement happen when iPads and game-based learning are incorporated into fifth-grade mathematics instruction?
*Journal of Information Technology Education: Research*,*11*, 269–286.Google Scholar - Clements, D. H., Battista, M. T., & Sarama, J. (2001). Logo and geometry.
*Journal for Research in Mathematics Education. Monograph,**10*, 1–177.CrossRefGoogle Scholar - Deterding, S., Dixon, D., Khaled, R., & Nacke, L. (2011). From game design elements to gamefulness: Defining “Gamification”.
*Communications of the ACM,**978*, 1–15.Google Scholar - Dorward, J. & Heal, R. (1999). National library of virtual manipulatives for elementary and middle level mathematics. In
*Proceedings of WebNet world conference on the WWW and internet 1999*(pp. 1510–1511). Association for the Advancement of Computing in Education (AACE), Chesapeake, VA. Retrieved August 12, 2015 from http://www.editlib.org/p/7655 - Goldin, G. A. (2003). Representation in school mathematics: A unifying research perspective. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.),
*A research companion to Principles and Standards for School Mathematics*(pp. 275–285). Reston, VA: NCTM.Google Scholar - Goldin, G., & Shteingold, N. (2001). Systems of representations and the development of mathematical concepts. In A. A. Cuoco & F. R. Curcio (Eds.),
*The roles of representation in school mathematics NCTM yearbook 2001*(pp. 1–23). Reston, VA: NCTM.Google Scholar - Haistings, J. L. (2009).
*Using virtual manipulatives with and without symbolic representation to teach first grade multi-digit addition*(Doctoral dissertation). Available from ProQuest Dissertations and Theses database (UMI No. 3366234).Google Scholar - Handal, B., & Herrington, A. (2003). Re-examining categories of computer-based learning in mathematics education.
*Contemporary Issues in Technology and Teacher Education,**3*(3), 275–287.Google Scholar - Heal, R., Dorward, J., & Cannon, L. (2002). Virtual manipulatives in mathematics: Addressing Conceptual dilemmas. In D. Willis, J. Price & N. Davis (Eds.),
*Proceedings of society for information technology and teacher education international conference 2002*(pp. 1056–1060). Association for the Advancement of Computing in Education (AACE), Chesapeake, VA.Google Scholar - Kaput, J. J. (1986). Information technology and mathematics: Opening new representational windows.
*The Journal of Mathematical Behavior,**5*(2), 187–207.Google Scholar - Kay, R. H. (2012). Examining factors that influence the effectiveness of learning objects in mathematics classrooms.
*Canadian Journal of Science, Mathematics, and Technology Education,**12*(4), 350–366.CrossRefGoogle Scholar - Kirby, K. D. (2013, April).
*The development of an idealized number line: Differentiating physical inscription from mathematical object.*Paper presented at the American Educational Research Association, San Francisco, CA.Google Scholar - Kurz, T. L., Middleton, J. A., & Yanik, H. B. (2005). A taxonomy of software for mathematics instruction.
*Contemporary Issues in Technology and Mathematics Teacher Education,**5*(2), 1–13.Google Scholar - Lazonder, A. W., & Ehrenhard, S. (2013). Relative effectiveness of physical and virtual manipulatives for conceptual change in science: How falling objects fall.
*Journal of Computer Assisted learning,**30*(2), 110–120.CrossRefGoogle Scholar - Manches, A., & O’Malley, C. (2012). Tangibles for learning: A representational analysis of physical manipulation.
*Personal and Ubiquitous Computing,**16*(4), 405–419.CrossRefGoogle Scholar - Martin, T., & Schwartz, D. L. (2005). Physically distributed learning: Restructuring and reinterpreting physical environments in the development of fraction concepts.
*Cognitive Science,**29*, 587–625.CrossRefGoogle Scholar - Moyer, P. S., Bolyard, J. J., & Spikell, M. A. (2002). What are virtual manipulatives?
*Teaching Children Mathematics,**8*(6), 372–377.Google Scholar - Moyer, P. S., Niezgoda, D., & Stanley, J. (2005). Young children’s use of virtual manipulatives and other forms of mathematical representations. In W. J. Masalski & P. C. Elliott (Eds.),
*Technology-supported mathematics learning environments: Sixty-seventh yearbook*(pp. 17–34). Reston, VA: NCTM.Google Scholar - Moyer-Packenham, P. S., & Suh, J. M. (2012). Learning mathematics with technology: The influence of virtual manipulatives on different achievement groups.
*Journal of Computers in Mathematics and Science Teaching,**31*(1), 39–59.Google Scholar - Moyer-Packenham, P. S., & Westenskow, A. (2013). Effects of virtual manipulatives on student achievement and mathematics learning.
*International Journal of Virtual and Personal Learning Environments,**4*(3), 35–50.CrossRefGoogle Scholar - Olympiou, G., & Zacharia, Z. C. (2012). Blending physical and virtual manipulatives: An effort to improve students’ conceptual understanding through science laboratory experimentation.
*Science Education,**96*(1), 21–47.CrossRefGoogle Scholar - Pea, R. D. (1985). Beyond amplification: Using the computer to reorganize mental functioning.
*Educational Psychologist,**20*, 167–182.CrossRefGoogle Scholar - 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,**24*(1), 5–25.Google Scholar - Resnick, M., Martin, F., Berg, R., Borovoy, R., Colella, V., Kramer, K., et al. (1998).
*Digital manipulatives: New toys to think with*. Proceedings of the SIGCHI conference on human factors in computing systems (pp. 18–23). MIT Media Laboratory, Boston, MA.Google Scholar - Riconscente, M. M. (2013). Results from a controlled study of the iPad fractions game motion math.
*Games and Culture*,*8*(4), 186–214. http://doi.org/10.1177/1555412013496894 Google Scholar - Sarama, J., & Clements, D. H. (2009). “Concrete” computer manipulatives in mathematics education.
*Child Development Perspectives,**3*(3), 145–150.CrossRefGoogle Scholar - Sedig, K., & Liang, H.-N. (2006). Interactivity of visual mathematical representations: Factors affecting learning and cognitive processes.
*Journal of Interactive Learning Research,**17*(2), 179–212.Google Scholar - Simon, M. A. (2013). The need for theories of conceptual learning and teaching of mathematics. In K. R. Leatham (Ed.),
*Vital directions for mathematics education research*(pp. 95–118). New York: Springer.Google Scholar - Steen, K., Brooks, D., & Lyon, T. (2006). The impact of virtual manipulatives on first grade geometry instruction and learning.
*Journal of Computers in Mathematics and Science Teaching,**25*(4), 373–391.Google Scholar - Suh, J., & Moyer, P. S. (2007). Developing students’ representational fluency using virtual and physical algebra balances.
*Journal of Computers in Mathematics and Science Teaching,**26*(2), 155–173.Google Scholar - Triona, L. M., & Klahr, D. (2003). Point and click or grab and heft: Comparing the influence of physical and virtual instructional materials on elementary school students’ ability to design experiments.
*Cognition and Instruction,**21*(2), 149–173.CrossRefGoogle Scholar - Tucker, S. I. (2015).
*An exploratory study of attributes, affordances, abilities, and distance in children’s use of mathematics virtual manipulative iPad apps*(Doctoral dissertation). ProQuest Dissertations and Theses database.Google Scholar - Wight, H., & Kitchenham, A. (2015). Virtual 10 frames and mobile technology in a Canadian primary classroom. In
*Mobile learning and mathematics*(pp. 135–149). New York, NY: Routledge.Google Scholar - Zacharia, Z. C., & deJong, T. (2014). The effects on students’ conceptual understanding of electric circuits of introducing virtual manipulatives within a physical manipulatives-oriented curriculum.
*Cognition and Instruction,**32*(2), 101–158.CrossRefGoogle Scholar