Quantitative Skills Vocabulary

In order to classify teaching materials (and other resources) according to the quantitative skills they address a common vocabulary is needed. The following preliminary vocabulary developed by Sam Patterson at Carleton College attempts to fufill this need.

We'd like to encourage you to you to join the quantitative skills email listserv and discuss how this vocabulary meets your needs. Does it cover the quantitative skills you think are important? Is it sufficiently detailed to allow accurate descriptions and precise searching? Is it sufficiently concise to be easy to work with? Are there regroupings or expansions of these terms that would make the vocabulary more useful?

Basic Skills

• Arithmetic -- Fractions, percentage, scaling
• Units -- Conversion, multiplication
• Scientific Notation -- Significant digits
• Simple Equations -- Solving for a single variable, quadratic formula
• Exponents -- Laws of exponents, relation to logarithms
• Logarithms -- Properties of logarithms, relation to exponents
• Complex numbers -- polar and exponential representations, arithmetic, Fundamental theorem of algebra
• Curve Fitting
• Optimization -- with and without constraints, with and without calculus
• Proof -- Mathematical reasoning, rigor, elementary logic

Geometry and Trigonometry

• Lines -- Equations, slope
• Planes -- Equations
• Area and volume of simple regions
• Conic sections -- Ellipse, parabola, hyperbola
• Quadric surfaces -- Ellipsoid, paraboloid, hyperboloid
• Trigonometry -- Relationships between length and angle
• Pythagorean theorem
• Regular solids -- Platonic, Archimedean
• Symmetry
• Coordinate systems -- Rectangular, polar, cylindrical, spherical, transformations
• 3-D -- Visualization, representation, slices, projections, perspective
• Dimension -- Conceptualizing higher dimensions

Graphs

• Interpretation of graphical information -- Draw inference
• Visualization of data -- Histograms, pie charts, scatter plots
• Log-log and log-linear plots
• Schematics -- Flowcharts, decision trees, concept maps
• Relationships among variables -- Concept of function
• Asymptotes

Functions

• Linear -- Slopes and intercepts
• Polynomial -- Factors and roots
• Rational -- Fractions of polynomials
• Power -- x^p, fractional powers, n^th roots
• Exponential -- p^x, growth/decay, relation to logarithm
• Logarithm -- Natural and base b, growth rate, relation to exponential
• Trigonometric -- Graphs, identities (periodicity, addition formulae, Pythagorean)

Calculus and Analysis

• Concept of function -- Addition, composition, invertibility
• Derivative -- Slope, rate of change, local properties of curves, computations
• Integral -- Area, volume, accumulation of small changes, computations
• Partial derivatives -- Functions of several variables, tangent planes, computations
• Gradients
• Level sets and contours and surfaces
• Multiple integrals -- Double, triple,...
• Line and surface integrals
• Vector fields -- Flux, divergence, curl, line and surface integrals
• Approximations of functions -- n^th order, Taylor and Fourier polynomials
• Ordinary differential equations -- Analytic and numerical methods, oscillators
• Partial differential equations -- Diffusion and wave equations
• Series -- Real, power, Taylor, Fourier
• Fourier Series and Transforms
• Spectral analysis

Linear Mathematics

• Vectors -- Visualization, arithmetic, dot-product, cross-product
• Systems of equations -- Graphical interpretation, matrix representation
• Linear Transformations -- Shear, scaling, rotation, reflection, projection
• Matrices -- Arithmetic, rank, transpose, inverse, determinant, sparse
• Eigenvalues and eigenvectors -- Symmetric matrices, adapted coordinates, principle components
• Orthogonality -- Projection, least squares, inner-product

Probability and Statistics

• Probability
• Exploratory Data Analysis -- Informal, qualitative assessment
• Design of experiments
• Sampling
• Hypothesis testing
• Distributions
• Sampling distributions
• Random Variables
• Regression
• Analysis of Variance
• Markov Chains
• Non-parametric statistics
• Spatial statistics

Interdisciplinary Concepts

• Mechanics -- Newton's laws, force, energy, work, oscillators
• Waves -- Wavelength, period, amplitude, frequency
• Simulation -- Deterministic, stochastic, numerical
• Numerical methods -- Interpolation, iteration, numerical linear algebra and differential equations
• Optimization
• Group theory

Technology

• Graphing calculators
• Elementary programming -- C, C++, Java, Pascal
• Mathematical software -- Mathematica, Maple, MatLab, Stella
• Stastisitcal software -- SPSS, S-PLUS, SAS, Minitab, Fathom
• Spreadsheets

Higher Order Skills

• Problem solving -- Formulate, Solve, and Interpret
• Problem solving strategies -- Explicit problem solving strategies
• Models -- Formulate, analyze, predict, linearity, non-linearity
• Order of magnitude calculations
• Estimation -- Reasonableness of results, checking answers
• Reading -- Comprehend and analyze mathematical text
• Writing -- Express quantitative ideas and facts effectively in writing
• Speaking -- Express quantitative ideas and facts effectively orally
• Information literacy -- Access and make effective use of quantitative information