2. Tech-knowledgy in Mathematics: Investigating Mathematical Models and Concepts Using Virtual Manipulatives and Applets Jennifer M. Suh, Ph.D. jsuh4@gmu.edu Assistant Professor of Mathematics Education George Mason University Christopher J. Johnston, M.A. cjohnst2@gmu.edu Ph.D. Student – Mathematics Education & Leadership (Instructional Technology, Secondary Concentration) George Mason University jsuh4@gmu.edu cjohnst2@gmu.edu Presentation at the National Education Computing Conference – Atlanta – June, 2007

3. Agenda Guidelines and research on appropriate uses of technology in mathematics Highlight some exemplary applets and investigations with Virtual Manipulatives Discuss instructional design and assessment Q and A

4. Are you… Classroom teacher? College or university faculty? Technology specialist or coordinator? Curriculum Specialists? Instructional designers? Others?

5. Appropriate Uses of Technology in Mathematics Education Five guidelines for appropriate uses of technology were identified by Garofalo, Drier, Harper, Timmerman, and Shockey (2000) and are specific to mathematics education: Introduce technology in context Address worthwhile mathematics with appropriate pedagogy Take advantage of technology Connect mathematics topics Incorporate multiple representations (p. 67).

6. REPRESENTATIONS-Defining teachers and students’ use Both presentational model (used by adults in instruction) Re-presentational model ( produced by students in learning) (Lamon, 2001)

7. “Models of thinking into models for thinking ” (Gravemeijer 1999, 2000) Helps students build a network of mathematical relationships and make generalizations Gives access to students to form mental images of concepts and context Importance of mathematical models

8. …from tools to represent thinking into models for thinking

9. …build a network of mathematical relationships

10. …build deeper understanding through multiple representations & mental images Example of Mathematical Model to teach density of rational numbers

11. Mathematical Models and Technology Effective Representational Models have… 1)Transparency: how easily can the idea be seen through the representation 2)Efficiency: Does the representation support efficient communication and use? 3)Generality: Does the representation apply to broad classes of objects or concepts? 4)Clarity: Is the representation unambiguous and easy to use 5)Precision: How close it the representation to the exact value? From National Research Council (2002) Adding it Up

12. Modeling perimeter and area http://www.shodor.org/interactivate1.0/activities/perm/index.htmlhttp://nlvm.usu.edu/en/nav/frames_asid_281_g_2_t_4.html?open =activities http://www.funbrain.com/cgi-bin/poly.cgi http://www.mathplayground.com/InteractiveGeometry.html

13. Modeling “ten-ness”

15. Designing lessons using VM Record thinking using a task sheet & print out work for assessment purposes Stress the importance of classroom discourse before, during and after using the tools Explore relationships and patterns in mathematics using tools Use the 5 criteria for selecting representations Use to conceptually understand procedural algorithm (ex. Fraction addition)

16. Affordances of Virtual Manipulatives and Applets Linked representations Immediate feedback Interactive and dynamic objects Non traditional methods to model math Differentiation

17. New (nontraditional) ways to model math ideas Probability “Law of Large Numbers”

18. Ease of differentiation and tiered learning

19. Assessment Issues when Using Technology Assessing the technology vs. assessing concepts learned via the technology Using a task sheet to promote reflection Math discussions: partners, whole-class Performance-based assessments Assessments which test transfer (without scaffolds)

20. Mathematical thinking Mathematical Thinking: –Understanding –Applying –Problem solving and reasoning –Making and testing conjectures –Creating

21. Mathematical Thinking

22. Sources of Virtual Manipulatives and Applets http://matti.usu.edu/nlvm/nav/vlibrary.html http://illuminations.nctm.org/tools/index.aspx http://www.shodor.org/interactivate/ http://mason.gmu.edu/~jsuh4/mathbridges/index.h tmlhttp://mason.gmu.edu/~jsuh4/mathbridges/index.h tml

23. References Battey, D., Kafai, Y, & Franke, M. (2005). Evaluation of mathematical inquiry in commercial rational number software. In C. Vrasidas & G. Glass (Eds.), Preparing teachers to teach with technology (pp. 241-256). Greenwich, CT: Information Age Publishing. Garofalo, J., Drier, H., Harper, S., Timmerman, M. A., & Shockey, T. (2000). Promoting appropriate uses of technology in mathematics teacher preparation. Contemporary Issues in Technology and Technology Education, 1(1), 66-88. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. http://standards.nctm.org/document/eexamples/index.htm http://standards.nctm.org/document/eexamples/index.htm National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

24. Q&A