Fostering HigherOrder Mathematical Thinking Using Technology

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Fostering Higher-Order Mathematical Thinking
Using Technology
Lingguo Bu lb04f_at_fsu.edu Rob Schoen
rschoen_at_zeno.math.fsu.edu Faculty Advisor Dr.
Maria Fernandez Florida State University Februar
y 25, 2005
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What does research say?

• Harold Wenglinskys findings
• Eighth graders whose teachers used computers
  mostly for simulations and applications--general
  ly associated with higher-order
  thinking--performed better on NAEP than students
  whose teachers did not. Meanwhile, 8th graders
  whose teachers used computers primarily for
  drill and practice--generally associated with
  lower-order thinking--performed worse.
• (Education Week on the Web, retrieved January
  7, 2005).
• Refer also to the National Educational Technology
  Plan.

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Learning Theory Bruners Three Modes of
Cognitive Representation

• Enactive (motor responses)
• Iconic (images)
• Symbolic (language, mathematical notations,
  computational models)

Adults and kids alike learn best using
representations appropriate for their age and,
most importantly, their level of cognitive
development.
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Mathematical Values of Sierpinskis Triangle

• Geometrically, it provides opportunity for
  learners to explore symmetry, similarity,
  transformation, etc.
• Algebraically, it is rich in patterns, relations,
  and measurement.
• Cognitively, it affords a connection between
  geometry and algebra, generating
  thought-provoking discourse in the classroom.
• Computationally, it invites reflections and
  fosters learner ownership.

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Multiple Approaches

• Lecture Lets talk about it.
• Hand-on Project Lets Cut.
• Paintbrush Lets draw on the computer.
• Computation Lets try programming.

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The Lecture Approach

• Instructor-centered presentation
• Little or no learner participation
• Little interaction between learners and the
  instructor.
• Result knowledge with little understanding or
  chance for knowledge transfer.
• Benefit time management and faster coverage of
  the lesson text.

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Paper and Scissors

• We observed two types of construction building
  and removing.
• Benefits engages learners, provides hand-on
  reference, generates discourse
• Drawback instruction time and lower-order
  understanding.

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The Paintbrush Approach

• Using the painting tool, we observed two types of
  behavior representing two types of understanding.
• Replication (Copy Paste)
• representing a partial view of the fractal as an
  outgrowing self-similar figure.
• Recursive Division
• representing a partial view of the fractal as an
  evolving structure on a microscopic scale.

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Paintbrush Replication(copypaste)
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Paintbrush Recursive Division
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A Computational Approach

• To Draw a Sierpinski Triangle
• if it is the deepest level, draw a filled
  triangle
• otherwise
• Draw a sierpinski triangle in the lower-left.
• Draw a sierpinski triangle at the top
• Draw a sierpinski triangle in the lower-right.

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A Mathematica Implementation
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A Computational Approach

• Benefits
• Provides a scaffold for learners to explore the
  various properties of the fractal toward an
  in-depth understanding of the mathematics behind.
• Helps develop an appreciation of mathematical
  rules.
• Generates a more detailed visualization of the
  structure.
• Enhance motivation for learners to take
  initiative in their mathematical experience.
• Drawbacks
• Software and hardware support.
• Experience in coding.
• Teachers knowledge of the integration of
  mathematics and computing and related pedagogical
  issues.

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Recommendations for Instruction

• Hands-on projects and computer-assisted
  exploration provide the foundation for learning
  cognitive development their roles in the
  classroom are not to be overlooked as learning
  aids.
• Instruction might follow the sequence of
  ethnomathematical knowledge, to intuitive, to
  technical symbolic, and axiomatic deductive
  knowledge (Kieren, T. E. 1993).
• Technology, especially, the computer, could be
  used at various levels of the instruction to
  facilitate learners visualization and
  exploration. In particular, computer simulation
  and learner-initiated programming contributes to
  higher-order mathematical thinking.

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Acknowledgement

• Work funded in part by Florida Department of
  Education in its STTA Grant (Students
  Transitioning Toward Algebra), directed Dr Maria
  Fernandez and Dr. Monica Hurdal.
• References
• Archer, J. (2005). The link to higher scores.
  Education Week on the Web.
• Bransford, John D., Ann L. Brown and Rodney
  R. Cocking (ed). (1999). How People Learn
  Brain, Mind, Experience, and School. Washington,
  D.C. National Academy Press.
• Driscoll, M. P. (2000) . Psychology of learning
  for instruction. (2nd ed.). Boston, MA Allyn
  and Bacon.
• Sowder, J. T. et al. (1998). Middle-grade
  teachers mathematical knowledge and its
  relationship to instruction. Albany, NY SUNY
  Press.