ìCoreî Subject Taxonomy for Mathematical Sciences Education

"Final" Draft

Proposed by the Mathematical Sciences Conference Group on Digital Educational Resources

February 2002

 

Brief Background:

A number of collections and organizations with a strong interest in using technology to enhance mathematical science education have worked to develop the "core" subject taxonomy for mathematical science education presented here.

 

Primary Goal:

…         To create a subject taxonomy for the mathematical sciences to which each participating digital library/collection is able to map their internal subject taxonomy.

 

Secondary Goals:

…         No more than approximately 9 topics per level (for human browsability)

…         End-user may use this taxonomy to catalog their resource

…         This taxonomy may be implemented in a browse structure

…         A digital library/collection may implement this as their vocabulary

…         Arbitrary levels of detail are allowed below the tree structure indicated

…         Changes to this taxonomy can be proposed and will follow a given easy revision process

 

Ground Rules for editing this document:

…         Add two asterisks, your email address and the date after a change in the hierarchy.

…         Record a description of your changes in the "Record of Changes" section, including your email and date of making the change.

 

(Note: This document was created in Microsoft Word and saved as a HTML/Web document.)

 

Record of Changes:

…         Moved ElementaryMathematics.Irrational.Algebraic, ElementaryMathematics.Irrational.pi and ElementaryMathematics.Irrational.e up one level in the taxonomy. (Working Group, 2/15/02)

…         Made new category Discrete Mathematics, put after Algebra and Number Theory (Working Group, 2/15/02)

…         Renamed and Algebra and Discrete Mathematics to Algebra and Number Theory (Working Group, 2/15/02)

…         Added DiscreteMathematics.Chaos (Working Group, 2/15/02)

…         Moved ElementaryMathematics.Data to StatisticsAndProbability.Data (Working Group, 2/15/02)

…         Added Geometry.FractalGeometry (Working Group, 2/15/02)

…         Change DifferentialEquations to DifferentialAndDifferenceEquations (Working Group, 2/15/02)

…         Changed DifferentialEquation.DynamicSystems.DifferentialDynamics to DifferentialEquation.DynamicSystems..DifferentiableDynamics (Working Group, 2/15/02)

…         Removed Analysis.NumericalAnalysis.Miscellaneous (Working Group, 2/15/02)

…         Added MathematicsHistory.General (Working Group, 2/15/02)

 

"Core" Subject Taxonomy for Mathematical Sciences Education

"Final" Draft

Proposed by the Mathematical Sciences Conference Group on Digital Educational Resources

February 2002

 

 

1.0      Elementary Mathematics

1.1      Numbers

1.1.1          Natural

1.1.2          Integers

1.1.3          Rational

1.1.4          Irrational

1.1.5          Algebraic

1.1.6          π

1.1.7          e

1.1.8          Real

1.1.9          Complex

1.2      Arithmetic

1.2.1           Operations

1.2.1.1          Addition

1.2.1.2          Subtraction

1.2.1.3          Multiplication

1.2.1.4          Division

1.2.1.5          Square Roots

1.2.1.6          Factorials

1.2.2           Fractions

1.2.2.1          Equivalent Fractions

1.2.2.2          Addition

1.2.2.3          Subtraction

1.2.2.4          Multiplication

1.2.2.5          Division

1.2.2.6          Ratio and Proportion

1.2.3           Decimals

1.2.3.1          Addition

1.2.3.2          Subtraction

1.2.3.3          Multiplication

1.2.3.4          Division

1.2.3.5          Percents

1.2.4           Estimation

1.2.5           Comparison of numbers

1.2.6           Exponents

1.2.6.1          Multiplication

1.2.6.2          Division

1.2.6.3          Powers

1.3 Patterns and Sequences

1.3.1           Geometric Patterns

1.3.2           Tilings and Tessellations

1.3.3           Golden Ratio

1.3.4           Fibonacci Sequence

1.3.5           Arithmetic Sequence

1.3.6           Geometric Sequence

1.4 Shapes and Figures

1.4.1           Plane shapes

1.4.2           Spatial Sense

1.4.3           Symmetry

1.4.4           Similar Figures

1.4.5           Solid Shapes

1.5 Measurement

1.5.1           Units of Measurement

1.5.1.1          Metric System

1.5.1.2          Standard Units

1.5.2           Linear Measure

1.5.2.1          Distance

1.5.2.2          Circumference

1.5.2.3          Perimeter

1.5.2.4          Scale

1.5.3           Area

1.5.3.1          Area of Polygons

1.5.3.2          Area of Circles

1.5.3.3          Surface Area

1.5.4           Volume

2.0     Logic and Foundations

2.1      Logic

2.1.1          Venn Diagrams

2.1.2          Propositional and Predicate Logic

2.1.3          Induction

2.1.4           Methods of Proof

2.2 Set Theory

2.2.1           Sets and Set Operations

2.2.2           Relations and Functions

2.2.3           Cardinality

2.2.4           Axiom of Choice

2.3 Computability, Decidability and Recursion

2.4 Model Theory

3.0     Algebra and Number Theory

3.1      Algebra

3.1.1           Graphing

3.1.2           Functions

3.1.2.1        Linear

3.1.2.2        Quadratic

3.1.2.3        Polynomial

3.1.2.4        Rational

3.1.2.5        Exponential

3.1.2.6        Logarithmic

3.1.2.7        Piece-wise

3.1.2.8        Step

3.1.3           Equations

3.1.3.1        Linear

3.1.3.2        Quadratic

3.1.3.3        Polynomial

3.1.3.4        Rational

3.1.3.5        Exponential

3.1.3.6        Logarithmic

3.1.3.7        Systems

3.1.4          Inequalities

3.1.5          Matrices

3.1.6          Sequences and Series

3.1.7          Algebraic Proof

3.2 Linear Algebra

3.2.1           Systems of Linear Equations

3.2.2           Matrix algebra

3.2.3           Vectors in R³

3.2.4           Vector Spaces

3.2.5           Linear Transformations

3.2.6           Eigenvalues and Eigenvectors

3.2.7           Inner Product Spaces

3.3 Abstract Algebra

3.3.1           Groups

3.3.2           Rings and Ideals

3.3.3           Fields

3.3.4           Galois Theory

3.3.5           Multilinear Algebra

3.4 Number Theory

3.4.1          Integers

3.4.2          Primes

3.4.2.1        Divisibility

3.4.2.2        Factorization

3.4.2.3        Distributions of Primes

3.4.3           Congruences

3.4.4           Diophantine Equations

3.4.5           Irrational Numbers

3.4.6           Famous Problems

3.4.7           Coding Theory

3.4.8           Cryptography

3.5      Category Theory

3.6      K-Theory

3.7      Homological Algebra

3.8      Modular Arithmetic

4.0     Discrete Mathematics

4.1      Cellular Automata

4.2      Combinatorics

4.3      Game Theory

4.4      Algorithms

4.5      Graph Theory

4.6      Linear Programming

4.7      Order and Lattices

4.8      Theory of Computation

4.9      Chaos

5.0     Geometry

5.1      Plane Geometry

5.1.1           Measurement

5.1.2           Geometric Proof

5.1.3           Parallel and Perpendicular Lines

5.1.4           Angles

5.1.5           Triangles

5.1.5.1        Pythagorean Theorem

5.1.5.2        Properties of Right Triangles

5.1.6           Congruence

5.1.7           Similarity

5.1.8           Polygons

5.1.8.1        Rectangles

5.1.8.2        Squares

5.1.8.3        Trapezoids

5.1.8.4        Pentagons

5.1.8.5        Hexagons

5.1.8.6        Regular Polygons

5.1.9           Circles

5.2 Solid Geometry

5.2.1          Lines and Planes

5.2.2          Angles

5.2.3          Spheres

5.2.4          Cones

5.2.5          Cylinders

5.2.6          Pyramids

5.2.7          Prisms

5.2.8          Polyhedra

5.3      Analytic Geometry

5.3.1          Cartesian Coordinates

5.3.2          Lines

5.3.3          Circles

5.3.4          Planes

5.3.5          Conics

5.3.6          Polar Coordinates

5.3.7          Parametric Curves

5.3.8          Surfaces

5.3.9          Curvilinear Coordinates

5.3.10      Distance Formula

5.4 Projective Geometry

5.5 Differential Geometry

5.6 Algebraic Geometry

5.7 Topology

5.7.1          Point Set Topology

5.7.2          General Topology

5.7.3          Differential Topology

5.7.4          Algebraic Topology

5.8 Trigonometry

5.8.1           Angles

5.8.2           Trigonometric Functions

5.8.3           Inverse Trigonometric Functions

5.8.4           Trigonometric Identities

5.8.5           Trigonometric Equations

5.8.6           Roots of Unity

5.8.7           Spherical Trigonometry

5.9 Fractal Geometry

6.0      Calculus

6.1      Single Variable

6.1.1           Functions

6.1.2           Limits

6.1.3           Continuity

6.1.4           Differentiation

6.1.5           Integration

6.1.6           Series

6.2      Several Variables

6.2.1           Functions of Several Variables

6.2.2           Limits

6.2.3           Continuity

6.2.4           Partial Derivatives

6.2.5           Multiple integrals

6.2.6           Taylor Series

6.3 Advanced Calculus

6.3.1          Vector Valued Functions

6.3.2          Line Integrals

6.3.3          Surface Integrals

6.3.4          Stokes Theorem

6.3.5          Linear spaces

6.3.6          Fourier Series

6.3.7          Orthogonal Functions

6.4      Tensor Calculus

6.5      Calculus of Variations

6.6      Operational Calculus

7.0     Analysis

7.1      Real Analysis

7.1.1           Metric Spaces

7.1.2           Convergence

7.1.3           Continuity

7.1.4           Differentiation

7.1.5           Integration

7.1.6           Measure Theory

7.2      Complex Analysis

7.2.1           Convergence

7.2.2           Infinite Series

7.2.3           Analytic Functions

7.2.4           Integration

7.2.5           Contour Integrals

7.2.6           Conformal Mappings

7.2.7           Several Complex Variables

7.3      Numerical Analysis

7.3.1           Computer Arithmetic

7.3.2           Solutions of Equations

7.3.3           Solutions of Systems

7.3.4           Interpolation

7.3.5           Numerical Differentiation

7.3.6           Numerical Integration

7.3.7           Numerical Solutions of ODEs

7.3.8           Numerical Solutions of PDEs

7.4      Signal Analysis

7.4.1          Fourier Series

7.5      Fourier Transforms

7.5.1           Filters

7.5.2          Noise

7.5.3           Sampling Theory

7.5.4           Wavelet Analysis

7.5.5           Data Compression

7.5.6           Image Processing

7.6      Functional Analysis

7.6.1           Hilbert Spaces

7.6.2           Banach Spaces

7.6.3           Topological Spaces

7.6.4           Locally Convex Spaces

7.6.5           Bounded Operators

7.6.6           Spectral Theorem

7.6.7           Unbounded Operators

7.7      Harmonic Analysis

7.8      Global Analysis

8.0      Differential and Difference Equations

8.1      Ordinary Differential Equations

8.1.1          First Order

8.1.2          Second Order

8.1.3          Linear Oscillations

8.1.4          Nonlinear Oscillations

8.1.5          Systems of Differential Equations

8.1.6          Sturm - Liouville Problems

8.1.7          Special Functions

8.1.8          Power Series Methods

8.1.9          Laplace Transforms

8.2      Partial Differential Equations

8.2.1          First Order

8.2.2          Elliptic

8.2.3          Parabolic

8.2.4          Hyperbolic

8.2.5          Integral Transforms

8.2.6          Integral Equations

8.2.7          Potential Theory

8.2.8          Nonlinear Equations

8.2.9          Symmetries and Integrability

8.3      Difference Equations

8.3.1          First Order

8.3.2          Second Order

8.3.3          Linear Systems

8.3.4          Z-Transforms

8.3.5          Orthogonal Polynomials

8.4 Dynamical Systems

8.4.1           1D Maps

8.4.2           2D Maps

8.4.3           Lyapunov Exponents

8.4.4           Bifurcations

8.4.5           Fractals

8.4.6           Differentiable Dynamics

8.4.7           Conservative Dynamics

8.4.8           Chaos

8.4.9           Complex Dynamical Systems

9.0      Statistics and Probability

9.1      Statistics

9.1.1           Sampling

9.1.2           Expectation Value and Variance

9.1.3           Linear Regression

9.1.4           Nonlinear Regression

9.1.5           Queuing Theory

9.1.6           Bayesian Statistics

9.2 Probability

9.2.1           Brownian Motion

9.2.2           Random Variables

9.2.2.1          Discrete Distributions

9.2.2.2          Continuous Distributions

9.2.2.3          Expectation Value

9.2.3           Central Limit Theorem

9.2.4           Markov Chains

9.2.5           Probability Measures

9.2.6           Stochastic Processes

9.3      Data

9.3.1          Data Representation

9.3.1.1        Bar graph

9.3.1.2        Box-and-whiskers plot

9.3.1.3        Circle graph/pie graph

9.3.1.4        Graphing

9.3.1.5        Histogram

9.3.1.6        Line-of-best-fit

9.3.1.7        Line plot

9.3.1.8        Pictograph

9.3.1.9        Scatter plot

9.3.1.10    Stem-and-leaf plot

9.3.1.11    Table

9.3.2          Data Collection

9.3.2.1        Experiment

9.3.2.2        Hypothesis

9.3.2.3        Sampling

9.3.2.4        Survey

9.3.3          Data Analysis

9.3.3.1        Measures of Central Tendency

1.1.1.1.1.           Mean

1.1.1.1.2.           Median

1.1.1.1.3.           Mode

9.3.3.2        Correlation

9.3.3.3        Distribution

10.0   Applied Mathematics

10.1   Mathematical Physics

10.2   Mathematical Economics

10.3   Mathematical Biology

10.4   Mathematics for Business

10.5   Engineering Mathematics

10.6   Mathematical Sociology

10.7   Mathematics for Social Sciences

10.8   Mathematics for Computer Science

11.0   Mathematics History

11.1   General

11.2   Famous Problems

11.3   Biographies of Mathematicians