In this section, you will find a number of different “mini” papers, covering a variety of things that we have learned about how students in upper-division learn. Most of these are based on action research—i.e. what we have observed in our classrooms and for which we have found some consistent evidence. Most of these topics would benefit from more systematic PER study. Please contact us if you would like to collaborate on a project!

our philosophy

  • connecting to mathematics that students DO know
  • “I just can't get started”
  • emphasizing geometric reasoning
  • supporting the development of harmonic reasoning
  • connecting chunks vs isolated facts

content organization

  • unifying concepts
  • using drvector to unify all of vector calculus
  • starting quantum mechanics with postulates
  • spins first
  • potentials first
  • paradigms vs. capstones (see video 07/12/19)

classroom techniques

  • creating the teachable moment
  • small groups
  • how to effectively run small groups
  • compare and contrast activities
  • room set-up
  • techniques for visualization activities
  • small white boards
  • kinesthetic activities
  • the roles of storytelling in the classroom
  • cementing the Ah Ha! moment with laughter

problems with notation

  • conventions for spherical coordinates
  • vectors
  • what sort of a “beast” is it?
  • dimensions–Shermer–searching for algebra error via dimension
  • is zero a number?


  • breaking a hard problem up into smaller pieces
  • encouraging the use of multiple representations

surprising things that students don't yet know

  • basis vectors that go with curvilinear coordinates
  • matrices as transformations
  • power series as approximations
  • eigenvectors as the things that are “unchanged” by transformation
  • a geometric conception of fields
  • that zero is a number
  • how to add two functions pointwise
  • how to read equations as words
  • what is planar about plane waves?
  • geometric interpretations or dot and cross products
  • the meaning of the vertical axis on a graph
  • how to shift a graph left or right, up or down

How Departments Change

  • Talk more with Jim Minstrell
  • Talk to faculty in biology at Maryland–how did they decide they wanted to change?
  • Lower Division change within Oregon State U. Physics Lower Division Change

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