The use of any understanding of the geometry of a problem in order to facilitate understanding of the problem and/or make progress on the problem.

Examples:

• Finding charge density on a ring applying the relationship that length = 2πR

• Converting between coordinate systems by using the geometry (not just using some previously established symbolic relationship)

• Finding the relationship between the φ-hat unit vector and the angle φ.

• Using the orthogonal relationship between gradient and equipotential surfaces to draw field lines

• Establishing that z = 0 by knowing something lies in the x-y plane.

• Using the geometry to establish which variables are held constant during integration (e.g. “we know the radius is the same at all angles so R is constant”)

• A drawing, gesture, verbal description, or manipulation of a physical object representing the shape or geometry of something

• A variable, expression, equation, or other mathematical representation that is not a picture. This can be either written or spoken in words or mathematical symbols.

• Starting with a symbolic representation such as an expression with numbers and variables, and creating a geometric representation such as a drawing or gesture that shows geometric understanding of the symbols

• Taking a drawing, gesture, or verbal description of the geometry of a problem and using it to produce an expression or equation

• Going back and forth between geometry and symbols such as symbols to picture to altering or manipulation of symbols, OR, drawing to symbols to altering of drawing, OR, pointing to symbols and drawing back and forth to establish the meaning of both (i.e how the drawing is representing the symbols and how the symbols reflect something in the drawing)

start with symbols and use some geometric representation that results in altering or adding to the symbolic representation OR start with a geometric representation and use or refer to symbols to result in altering the geometric representation. NOTE: starting with several symbols and simply transferring the meaning of each of those to build a more complete drawing would be only symbolic to geometric translation, to be a cycle the geometric representation has to somehow impact the symbolic process and vice versa.

more than one consecutive cycle

use of both symbols and drawing to understand a concept – this is tricky to define, so I will give examples:

 A student pointing to a drawing and an equation simultaneously or back and forth in succession between them while verbally describing or arguing for a particular interpretation

 A student gesturing while saying the symbolic language to describe or argue a particular point