0011 0010 1010 1101 0001 0100 1011
1
2
4
Tech-knowledgy in Mathematics:
Investigating Mathematical Models and
Concepts Using Virtual Manipulatives and
Applets
1
Jennifer M. Suh, Ph.D.
jsuh4@gmu.edu
0011 0010 1010 1101 0001 0100
1011
Assistant
Professor of Mathematics Education
George Mason University
2
4
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
Agenda
0011 0010 1010 1101 0001 0100 1011
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
1
2
4
Are you…
0011 0010 1010 1101 0001 0100 1011
Classroom teacher?
College or university faculty?
Technology specialist or coordinator?
Curriculum Specialists?
Instructional designers?
Others?
1
2
4
Appropriate Uses of Technology in
Mathematics
Education
0011 0010 1010 1101 0001 0100 1011
Five guidelines for appropriate uses of technology
were identified by Garofalo, Drier, Harper,
Timmerman, and Shockey (2000) and are
specific to mathematics education:
•
•
•
•
•
1
Introduce technology in context
Address worthwhile mathematics with appropriate pedagogy
Take advantage of technology
Connect mathematics topics
Incorporate multiple representations (p. 67).
2
4
REPRESENTATIONS-Defining
teachers and students’ use
0011 0010 1010 1101 0001 0100 1011
Both presentational model
(used by adults in instruction)
Re-presentational model
( produced by students in learning)
(Lamon, 2001)
1
2
4
Importance of mathematical models
0011 0010 1010 1101 0001 0100 1011
• “Models of thinking into models for
thinking” (Gravemeijer 1999, 2000)
1
• Helps students build a network of
mathematical relationships and make
generalizations
2
4
• Gives access to students to form mental
images of concepts and context
…from tools to represent thinking
into models for thinking
0011 0010 1010 1101 0001 0100 1011
1
2
4
…build a network of
mathematical relationships
0011 0010 1010 1101 0001 0100 1011
1
2
4
…build deeper understanding through
multiple representations &
mental images
0011 0010 1010 1101 0001 0100 1011
1
2
4
Example of Mathematical Model to teach
density of rational numbers
Mathematical Models and Technology
0011 0010 1010 1101 0001 0100 1011
Effective Representational Models have…
1)
2)
3)
4)
5)
Transparency: how easily can the idea be seen through the
representation
Efficiency: Does the representation support efficient
communication and use?
Generality: Does the representation apply to broad classes of
objects or concepts?
Clarity: Is the representation unambiguous and easy to use
Precision: How close it the representation to the exact value?
1
2
4
From National Research Council (2002) Adding it Up
Modeling perimeter and area
0011 0010 1010 1101 0001 0100 1011
http://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
1
2
4
http://www.mathplayground.com/InteractiveGeometry.html
http://www.funbrain.com/cgi-bin/poly.cgi
Modeling “ten-ness”
0011 0010 1010 1101 0001 0100 1011
1
2
4
Modeling “ten-ness”
0011 0010 1010 1101 0001 0100 1011
1
2
4
Designing lessons using VM
0011 0010 1010 1101 0001 0100 1011
• 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)
1
2
4
Affordances of Virtual
Manipulatives and Applets
0011 0010 1010 1101 0001 0100 1011
Linked representations
Immediate feedback
Interactive and dynamic objects
Non traditional methods to model math
Differentiation
1
2
4
New (nontraditional) ways to model math ideas
Probability
“Law of Large Numbers”
0011 0010 1010 1101 0001 0100 1011
1
2
4
Ease of differentiation and tiered
learning
0011 0010 1010 1101 0001 0100 1011
1
2
4
Assessment Issues when Using
Technology
0011 0010 1010 1101 0001 0100 1011
• 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)
1
2
4
Mathematical thinking
0011 0010 1010 1101 0001 0100 1011
Mathematical Thinking:
–
–
–
–
–
Understanding
Applying
Problem solving and reasoning
Making and testing conjectures
Creating
1
2
4
Mathematical Thinking
0011 0010 1010 1101 0001 0100 1011
1
2
4
Sources of Virtual Manipulatives
and Applets
0011 0010 1010 1101 0001 0100 1011
•
•
•
•
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
tml
1
2
4
References
0011 0010 1010 1101 0001 0100 1011
•
•
•
•
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.
1
2
4
Q&A
0011 0010 1010 1101 0001 0100 1011
1
2
4