Tuesday, January 15, 2013

standards for multivariable calculus, first pass

OK, it’s time to get real with this. In my last post, I explained some of my reasons for attempting to use SBG this spring and listed seven general standards for college-level mathematics classes. Now I’m listing the standards I have created specifically for multivariable calculus. There’s still time to tweak these, so I would certainly appreciate any feedback over the next few days (or, indeed, at any time!).

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).
During the semester, the content on which a standard is based may be introduced gradually over time. For this reason as well as the general expectation that skills should remain honed, many of the standards will be assessed several times. This is also one of my main sources of concern for confusion—what does it mean to have “mastered differentiation operators” at the level of computing partial derivatives and gradients, but not curl and divergence? If I were to distinguish these standards further simply because some parts are separated temporally, however, the number would increase two- or three-fold, making grading an intractable problem for me.

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
Functions, curves, and surfaces
  • 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 integralseffectively parametrize curves and surfaces, and use these use 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
Classical theorems
  • 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
In my next post (probably tomorrow), I’ll explain how I plan to grade these, including what scale I’ll use, how assignments will be broken into their component standards, and how students may improve their score on an individual standard.

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