Title: Easing the Transition to Upper-Division E & M

Abstract: Why do strong students who have done well in lower-division mathematics classes suddenly have trouble applying that knowledge in upper-division physics? How can we help students develop the complex three-dimensional visualization resources that they need to solve upper-division E & M problems? How can we encourage students to make genuine symmetry arguments when employing Ampere's law, rather than just parroting the words “by symmetry, this is obvious”? We will report on upper-division curricular materials and active-engagement strategies, developed for the Paradigms in Physics Project at Oregon State University, that can be used in upper-division E & M and/or mathematical methods courses.

• Approximations
• r→|r-r'|→sqrt{r^2+r'^2+…}
• Superposition→ring sequence (integration)
• Densities M/V vs. linear, surface, vol. vs. current
• Curvilinear coordinates
• Gauss and Ampere
• Boundary conditions
• Derivatives (Div, grad, curl, and theorems)
• Field lines vs. vectors vs scalars→ quadrupole, can calculate

potential vs. thinking of scalar fields

• Constants and variables

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Answers as numbers vs. answers as algebraic expressions vs answers as elliptic integrals

Harmonic reasning-.repeatedly going back to the geometry (Len): drawing the picture and equation shopping is not enough

What do students know (somewhat reliably) vs. what have students never experienced→much of this they pick up by osmosis in a traditional curriculum

Far deeper mathematical approach than at the lower-division

lower-div: visceral understanding→simple mathematical formulas

Use non-Len examples that show a variety of pedagogical strategies, as well as what students do and don't know from lower-division, as well as sequencing, as well as classroom conversations.

We will need to populate lots of activities and also sequences of activities.

Point to Len's poster, particularly, and our other talks, portfolios wiki, Ampere paper, computational paper

Content Matters

• Show Pat Heller diagram–start out slowly helps more students get to good endpoints–still want good endpoints–not dumbing down curriculum
• Throw away the textbook–this is how we transmit curriculum–becoming standardized–should students see your own notes?
• Example: Visualization: 3 representations: electrostatic potentials, electric fields, electric field lines
  • From lower-division, students typically know electric field lines best, then potentials.
  • We start with potentials first.
    • Emphasize the field idea
      • problems with derivative as a function, relation to constants vs. variables.
      • gradient are easier to see than line integrals
    • Describe the phenomenon of chunking
    • If you try immediately to tie to something students know, it is hard to pull them away from that, they always use old tried and true reasoning.
    • We see students trying to bring in vector field reasoning inappropriately when asked to think about potentials, i.e. trying to argue that: contributions to the potential from the ring cancel, contributions on axis due to symmetric point charges yield zero potential.
  • Instead, build a robust, chunked understanding of potentials first, insist that students use potentials reasoning, then overlay the chunks
  • Electric field vectors are a more important representation than electric field lines if you want student to understand superposition, chopping and adding. Electric field lines are more important if you want students to understand conservation and symmetry.
  • What students do reliably: draw potentials perpendicular to field lines (although, if the field lines are wrong, they don't have reasoning skills with potentials to get them out of the hole–see Steve, John, Jeff group fro quadrupole activity.) They have reasonable field line representations for point charges and parallel plates.
  • What students don't do : think about spacing between equipotential surfaces.
• What is the sequence of activities that we do?
• Students will solve a problem in 2-dimensions unless forced to think about the third. dipole→ring –carrying around physical representations–hula hoop and volt meter–help with this.

• Plane waves, dot products, using multiple representations of do products,
• div, grad, curl geometric definitions of divergence and curl
• gradient including some Bridge activities
• making approximations
• chopping and adding