Enhancing Undergraduate Geometric Skills and Thinking Perspectives in Mathematics Education

1. Skills of GEOMETRIC THINKINGin undergraduate level Arash Rastegar Assistant Professor Sharif University of Technology

2. 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.

3. Doing mathematics has an important rolein 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.

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

5. 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

6. 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

7. 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

8. 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

9. Doing mathematics has an important rolein learning and development of mathematics. • Logical assumptions based on experience • Internalization • experience does not replace rigorous arguments • Analysis and comparison with others’

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

11. 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

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

13. 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

14. 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

15. Geometric Thinking

16. Geometric Imagination • Three dimensional intuition • Higher dimensional intuition • Combinatorial intuition • Creative imagination • Abstract imagination

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

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

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

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

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

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

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

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

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

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

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

28. Deep Inside Geometry

29. Differential geometry and differentiable manifolds • Geometric Modeling • Classical mechanics

30. History of mathematical concepts

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