Skills of GEOMETRIC THINKING in undergraduate level

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Skills of GEOMETRIC THINKINGin undergraduate
level

• Arash Rastegar
• Assistant Professor
• Sharif University of Technology

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Perspectives towards mathematics education

• Mathematics education has an important role in
  development of mental abilities.
• Mathematics education is an efficient tool in
  developing the culture of scientific curiosity.
• We use mathematics in in solving everyday
  problems.
• Development of the science of mathematics and our
  understanding of nature are correlated.

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• Doing mathematics has an important role in
  learning and development of mathematics.
• Development of tools and Technology is correlated
  with development of mathematics.
• We use mathematics in studying, designing, and
  evaluation of systems.
• We use mathematical modeling in solving everyday
  problems.

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• Group thinking and learning is more efficient
  than individual learning.
• Mathematics is a web of connected ideas, concepts
  and skills.

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Mathematics education has an important role in
development of mental abilities.

• Skills of communication
• Strategies of thinking
• Critical thinking
• Logical thinking
• Creative thinking
• Imagination
• Abstract thinking
• Symbolical thinking
• Stream of thinking

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Mathematics education is an efficient tool in
developing the culture of scientific curiosity.

• Critical character
• Accepting critical opinions
• Using available information
• Curiosity and asking good questions
• Rigorous description
• Comparison with results of other experts
• Logical assumptions
• Development of new theories

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We use mathematics in in solving everyday
problems.

• Common mathematical structures
• Designing new problems
• Scientific judgment
• Development of mathematics to solve new problems
• Mathematical modeling

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Development of the science of mathematics and our
understanding of nature are correlated.

• Getting ideas from nature
• Study of the nature
• Control of nature
• Nature chooses the simplest ways

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Doing mathematics has an important role in
learning and development of mathematics.

• Logical assumptions based on experience
• Internalization
• experience does not replace rigorous arguments
• Analysis and comparison with others

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Development of tools and Technology is correlated
with development of mathematics.

• Limitations of Technology
• Mathematical models affect technology
• Utilizing technology in education

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We use mathematics in studying, designing, and
evaluation of systems.

• Viewing natural and social phenomena as
  mathematical systems
• Division of systems to subsystems
• Similarities of systems
• Summarizing in a simpler system
• Analysis of systems
• Mathematical modeling is studying the systems
• Changing existing systems

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We use mathematical modeling in solving everyday
problems.

• Utilizing old models in similar problems
• Limitations of models
• Getting ideas from models
• finding simplest models

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Group thinking and learning is more efficient
than individual learning.

• Problems are solved more easily in groups
• Comparing different views
• Group work develops personal abilities
• Morals of group discussion

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Mathematics is a web of connected ideas, concepts
and skills.

• Solving a problem with different ideas
• Atlas of concepts and skills
• Webs help to discover new ideas
• Mathematics is like a tree growing both from
  roots and branches

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Geometric Thinking
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• Geometric Imagination
• Three dimensional intuition
• Higher dimensional intuition
• Combinatorial intuition
• Creative imagination
• Abstract imagination

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• Geometric Arguments
• Set theoretical arguments
• Local arguments
• Global arguments
• Superposition
• Algebraic coordinatization

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• Geometric Description
• Global descriptions
• Local descriptions
• Algebraic descriptions
• Combinatorial descriptions
• More abstract descriptions

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• Geometric Assumptions
• Local assumptions
• Global assumptions
• Algebraic assumptions
• Combinatorial assumptions
• More abstract assumptions

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• Recognition of Geometric Structures
• Set theoretical structures
• Local structures
• Global structures
• Algebraic structures
• Combinatorial structures

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• Mechanics (Geometric Systems)
• Material point mechanics
• Solid body mechanics
• Fluid mechanics
• Statistical mechanics
• Quantum mechanics

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• Construction of Geometric Structures
• Set theoretical structures
• Local structures
• Global structures
• Algebraic structures
• Combinatorial structures

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• Doing Geometry
• Algebraic calculations
• Limiting cases
• Extreme cases
• Translation between different representations
• Abstractization

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• Techno-geometer
• Drawing geometric objects by computer
• Algebrization of geometric structures
• Performing computations by computer
• Algorithmic thinking
• Producing software fit to specific problems

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• Geometric Modeling
• Linear modeling
• Algebraic modeling
• Exponential modeling
• Combinatorial modeling
• More abstract modeling

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• Geometric Categories
• Topological category
• Smooth category
• Algebraic category
• Finite category
• More abstract categories

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• Roots and Branches of Geometry
• Euclidean geometry
• Spherical and hyperbolic geometries
• Space-time geometry
• Manifold geometry
• Non-commutative geometry

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Deep Inside Geometry
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• Differential geometry and differentiable
  manifolds
• Geometric Modeling
• Classical mechanics

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• History of mathematical concepts

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• Number Theory
• is the queen of mathematics.
• Geometry
• is the king of number theory and mathematics!