I have grouped the standards into five larger categories: “geometry of vectors”, “functions, curves, and surfaces”, “differentiation”, “integration”, and “classical theorems”. In some SBG implementations, the competencies listed below might be more finely sorted out, but I have come to believe that giving college students a somewhat broader classification will encourage them to guide their own education and to think holistically about the material. I have tried to sort these according to three basic principles:
- they cover roughly the same amount of course material;
- they are roughly of the same importance towards mastering the content;
- they can be more-or-less independently measured (although there are indisputably dependences among them).
Despite these misgivings about the list itself, I feel it’s important to go ahead and publish it so that I and others can reflect on it. So here goes: (Update 1/18: Following some feedback and discussion, I have revised this list from its original form. Please ignore the parts that have been struck out.)
Geometry of vectors
- Operations – compute and interpret sum, scalar multiples, dot product, determinant, and cross product of vectors in the plane or in space
- Objects – describe points, lines, planes, spheres, and other surfaces using equations, vectors, set notation, or geometric objects of a different kind
- Visualization – sketch or predict appearance of the graph of a function or curve based on a formula or other description; sketch or describe level sets of a function; use computer software to examine shapes of graphs
- Parametrization (added) – find parametrizations of lines, circles and other curves, as well as planes, spheres, and other surfaces (e.g., tori and graphs of functions)
- Analysis – find domain and range of functions of 1, 2, and 3 variables, and describe these using set notation or geometric terminology; determine continuity
- Operators – apply and interpret partial derivative, Jacobian, and gradient operators to functions; divergence and curl to vector fields
- Operations on functions (added) – compute and interpret partial derivatives and gradient of a function of 2 or 3 variables; apply and explain equality of mixed partial derivatives, including sufficient conditions for such equality to hold
- Operations on vector fields (added) – compute and interpret divergence and curl of a vector field
- Linearization – find tangent vectors, tangent lines, and tangent planes; use these to approximate curves and surfaces near a point
- Higher derivatives – apply and explain equality of mixed partial derivatives; use higher derivatives to collect data about shape of the graph of a function, including classifying critical points
- Optimization (added) – use higher derivatives to collect data about shape of the graph of a function; classify critical points; Lagrange multipliers
- Differential equations – interpret partial differential equations (such as the wave equation, the heat equation, and the Laplace equation) in terms of their solutions; verify that a given function is a solution to a PDE; use a computer to solve or approximately solve PDEs
- Multiple integrals – accurately describe regions over which double or triple integrals are computed; perform calculations of double or triple integrals; apply and justify change-of-coordinate formulas; use computers to find integrals
- Line and surface integrals –
effectively parametrize curves and surfaces, and use theseuse parametrizations of curves and surfaces to compute length, area, work, and flux integrals - Applications – use double and triple integrals to solve problems in geometry and probability; use line and surface integrals to investigate physical phenomena
- Integrability conditions – check conditions for a vector field to be irrotational or divergence-free (“closed” with respect to curl or divergence operators), and explain the meaning of these conditions; find potential functions
- Generalizations of FTC – explain the meaning and significance of Green’s Theorem, Divergence Theorem, and Stokes’ Theorem
- Applications – use Green’s Theorem, the Divergence Theorem, and Stokes’ Theorem to convert integrals between various forms
- Generalizations of FTC – explain the meaning and significance of Green’s Theorem, Divergence Theorem, and Stokes’ Theorem; use these to convert integrals between various forms
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