Skills of Skills of GEOMETRIC THINKINGGEOMETRIC THINKING

in undergraduate levelin undergraduate level

Arash RastegarArash Rastegar

Assistant ProfessorAssistant Professor

Sharif University of TechnologySharif University of Technology

Perspectives towards Perspectives towards mathematics educationmathematics education

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

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

We use mathematics in in solving everyday We use mathematics in in solving everyday problems.problems.

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

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

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

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

We use mathematical modeling in solving We use mathematical modeling in solving everyday problems.everyday problems.

Group thinking and learning is more efficient Group thinking and learning is more efficient than individual learning.than individual learning.

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

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

Skills of communicationSkills of communication Strategies of thinkingStrategies of thinking Critical thinkingCritical thinking Logical thinkingLogical thinking Creative thinking Creative thinking ImaginationImagination Abstract thinkingAbstract thinking Symbolical thinkingSymbolical thinking Stream of thinkingStream of thinking

Mathematics education is an efficient Mathematics education is an efficient tool in developing the culture of tool in developing the culture of

scientific curiosity.scientific curiosity. Critical character Critical character Accepting critical opinionsAccepting critical opinions Using available informationUsing available information Curiosity and asking good questionsCuriosity and asking good questions Rigorous descriptionRigorous description Comparison with results of other expertsComparison with results of other experts Logical assumptionsLogical assumptions Development of new theoriesDevelopment of new theories

We use mathematics in in solving We use mathematics in in solving everyday problems.everyday problems.

Common mathematical structuresCommon mathematical structures Designing new problemsDesigning new problems Scientific judgmentScientific judgment Development of mathematics to solve new Development of mathematics to solve new

problemsproblems Mathematical modelingMathematical modeling

Development of the science of Development of the science of mathematics and our understanding of mathematics and our understanding of

nature are correlated.nature are correlated. Getting ideas from natureGetting ideas from nature Study of the natureStudy of the nature Control of natureControl of nature Nature chooses the simplest waysNature chooses the simplest ways

Doing mathematics has an important Doing mathematics has an important rolerole in learning and development of in learning and development of

mathematics.mathematics. Logical assumptions based on experienceLogical assumptions based on experience InternalizationInternalization experience does not replace rigorous argumentsexperience does not replace rigorous arguments Analysis and comparison with others’Analysis and comparison with others’

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

mathematics.mathematics. Limitations of TechnologyLimitations of Technology Mathematical models affect technologyMathematical models affect technology Utilizing technology in educationUtilizing technology in education

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

Viewing natural and social phenomena as Viewing natural and social phenomena as mathematical systemsmathematical systems

Division of systems to subsystemsDivision of systems to subsystems Similarities of systemsSimilarities of systems Summarizing in a simpler systemSummarizing in a simpler system Analysis of systemsAnalysis of systems Mathematical modeling is studying the systemsMathematical modeling is studying the systems Changing existing systemsChanging existing systems

We use mathematical modeling in We use mathematical modeling in solving everyday problems.solving everyday problems.

Utilizing old models in similar problemsUtilizing old models in similar problems Limitations of modelsLimitations of models Getting ideas from modelsGetting ideas from models finding simplest modelsfinding simplest models

Group thinking and learning is more Group thinking and learning is more efficient than individual learning.efficient than individual learning.

Problems are solved more easily in groupsProblems are solved more easily in groups Comparing different viewsComparing different views Group work develops personal abilitiesGroup work develops personal abilities Morals of group discussionMorals of group discussion

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

Solving a problem with different ideasSolving a problem with different ideas Atlas of concepts and skillsAtlas of concepts and skills Webs help to discover new ideasWebs help to discover new ideas Mathematics is like a tree growing both from Mathematics is like a tree growing both from

roots and branchesroots and branches

Geometric ThinkingGeometric Thinking

Geometric ImaginationGeometric Imagination

1.1. Three dimensional intuitionThree dimensional intuition

2.2. Higher dimensional intuitionHigher dimensional intuition

3.3. Combinatorial intuitionCombinatorial intuition

4.4. Creative imaginationCreative imagination

5.5. Abstract imaginationAbstract imagination

Geometric ArgumentsGeometric Arguments

1.1. Set theoretical argumentsSet theoretical arguments

2.2. Local argumentsLocal arguments

3.3. Global argumentsGlobal arguments

4.4. SuperpositionSuperposition

5.5. Algebraic coordinatizationAlgebraic coordinatization

Geometric DescriptionGeometric Description

1.1. Global descriptionsGlobal descriptions

2.2. Local descriptionsLocal descriptions

3.3. Algebraic descriptionsAlgebraic descriptions

4.4. Combinatorial descriptionsCombinatorial descriptions

5.5. More abstract descriptionsMore abstract descriptions

Geometric AssumptionsGeometric Assumptions

1.1. Local assumptions Local assumptions

2.2. Global assumptionsGlobal assumptions

3.3. Algebraic assumptionsAlgebraic assumptions

4.4. Combinatorial assumptionsCombinatorial assumptions

5.5. More abstract assumptionsMore abstract assumptions

Recognition of Geometric StructuresRecognition of Geometric Structures

1.1. Set theoretical structuresSet theoretical structures

2.2. Local structuresLocal structures

3.3. Global structuresGlobal structures

4.4. Algebraic structuresAlgebraic structures

5.5. Combinatorial structuresCombinatorial structures

Mechanics (Geometric Systems)Mechanics (Geometric Systems)

1.1. Material point mechanicsMaterial point mechanics

2.2. Solid body mechanicsSolid body mechanics

3.3. Fluid mechanicsFluid mechanics

4.4. Statistical mechanicsStatistical mechanics

5.5. Quantum mechanicsQuantum mechanics

Construction of Geometric StructuresConstruction of Geometric Structures

1.1. Set theoretical structuresSet theoretical structures

2.2. Local structuresLocal structures

3.3. Global structuresGlobal structures

4.4. Algebraic structuresAlgebraic structures

5.5. Combinatorial structuresCombinatorial structures

Doing GeometryDoing Geometry

1.1. Algebraic calculationsAlgebraic calculations

2.2. Limiting casesLimiting cases

3.3. Extreme casesExtreme cases

4.4. Translation between different representationsTranslation between different representations

5.5. AbstractizationAbstractization

Techno-geometerTechno-geometer

1.1. Drawing geometric objects by computerDrawing geometric objects by computer

2.2. Algebrization of geometric structuresAlgebrization of geometric structures

3.3. Performing computations by computerPerforming computations by computer

4.4. Algorithmic thinkingAlgorithmic thinking

5.5. Producing software fit to specific problemsProducing software fit to specific problems

Geometric ModelingGeometric Modeling

1.1. Linear modelingLinear modeling

2.2. Algebraic modelingAlgebraic modeling

3.3. Exponential modelingExponential modeling

4.4. Combinatorial modelingCombinatorial modeling

5.5. More abstract modelingMore abstract modeling

Geometric CategoriesGeometric Categories

1.1. Topological categoryTopological category

2.2. Smooth categorySmooth category

3.3. Algebraic categoryAlgebraic category

4.4. Finite categoryFinite category

5.5. More abstract categoriesMore abstract categories

Roots and Branches of GeometryRoots and Branches of Geometry

1.1. Euclidean geometryEuclidean geometry

2.2. Spherical and hyperbolic geometriesSpherical and hyperbolic geometries

3.3. Space-time geometrySpace-time geometry

4.4. Manifold geometryManifold geometry

5.5. Non-commutative geometryNon-commutative geometry

Deep Inside GeometryDeep Inside Geometry

Differential geometry and differentiable Differential geometry and differentiable manifoldsmanifolds

Geometric ModelingGeometric Modeling Classical mechanicsClassical mechanics

History of mathematical conceptsHistory of mathematical concepts

Number Theory Number Theory

is the queen of mathematics. is the queen of mathematics.

GeometryGeometry

is the king of number theory and mathematics!is the king of number theory and mathematics!