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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 EducationGeorge Mason University

Christopher J. Johnston, M.A.cjohnst2@gmu.edu

Ph.D. Student – Mathematics Education & Leadership (Instructional Technology, Secondary Concentration)George Mason University

Presentation at the National Education Computing Conference – Atlanta – June, 2007

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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

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Are you…

Classroom teacher?

College or university faculty?

Technology specialist or coordinator?

Curriculum Specialists?

Instructional designers?

Others?

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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).

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REPRESENTATIONS-Defining teachers and students’ use

Both presentational model

(used by adults in instruction)

Re-presentational model

( produced by students in learning)

(Lamon, 2001)

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• “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

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…from tools to represent thinking into models for thinking

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…build a network of mathematical relationships

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…build deeper understanding through multiple representations &

mental images

Example of Mathematical Model to teach density of rational numbers

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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 use5) Precision: How close it the representation to the exact value?

From National Research Council (2002) Adding it Up

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Modeling perimeter and areahttp://www.shodor.org/interactivate1.0/activities/perm/index.html

http://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

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Modeling “ten-ness”

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Modeling “ten-ness”

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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)

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Affordances of Virtual Manipulatives and Applets

Linked representations

Immediate feedback

Interactive and dynamic objects

Non traditional methods to model math

Differentiation

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New (nontraditional) ways to model math ideas

Probability

“Law of Large Numbers”

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Ease of differentiation and tiered learning

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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)

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Mathematical thinking

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

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Mathematical Thinking

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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.html

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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

• National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

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Q&A