Skills of GEOMETRIC THINKING in undergraduate level Arash Rastegar Assistant Professor Sharif University of Technology

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

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

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

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

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

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

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

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’

Development of tools and Technology is correlated with development of mathematics.  Limitations of Technology  Mathematical models affect technology  Utilizing technology in education

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

We use mathematical modeling in solving everyday problems.  Utilizing old models in similar problems  Limitations of models  Getting ideas from models  finding simplest models

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

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

Geometric Thinking

Geometric Imagination 1. Three dimensional intuition 2. Higher dimensional intuition 3. Combinatorial intuition 4. Creative imagination 5. Abstract imagination

Geometric Arguments 1. Set theoretical arguments 2. Local arguments 3. Global arguments 4. Superposition 5. Algebraic coordinatization

Geometric Description 1. Global descriptions 2. Local descriptions 3. Algebraic descriptions 4. Combinatorial descriptions 5. More abstract descriptions

Geometric Assumptions 1. Local assumptions 2. Global assumptions 3. Algebraic assumptions 4. Combinatorial assumptions 5. More abstract assumptions

Recognition of Geometric Structures 1. Set theoretical structures 2. Local structures 3. Global structures 4. Algebraic structures 5. Combinatorial structures

Mechanics (Geometric Systems) 1. Material point mechanics 2. Solid body mechanics 3. Fluid mechanics 4. Statistical mechanics 5. Quantum mechanics

Construction of Geometric Structures 1. Set theoretical structures 2. Local structures 3. Global structures 4. Algebraic structures 5. Combinatorial structures

Doing Geometry 1. Algebraic calculations 2. Limiting cases 3. Extreme cases 4. Translation between different representations 5. Abstractization

Techno-geometer 1. Drawing geometric objects by computer 2. Algebrization of geometric structures 3. Performing computations by computer 4. Algorithmic thinking 5. Producing software fit to specific problems

Geometric Modeling 1. Linear modeling 2. Algebraic modeling 3. Exponential modeling 4. Combinatorial modeling 5. More abstract modeling

Geometric Categories 1. Topological category 2. Smooth category 3. Algebraic category 4. Finite category 5. More abstract categories

Roots and Branches of Geometry 1. Euclidean geometry 2. Spherical and hyperbolic geometries 3. Space-time geometry 4. Manifold geometry 5. Non-commutative geometry

Deep Inside Geometry

Differential geometry and differentiable manifolds Differential geometry and differentiable manifolds Geometric Modeling Geometric Modeling Classical mechanics Classical mechanics

History of mathematical concepts History of mathematical concepts

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