Teaching With Projections in the Geosciences

Geoscientists are trained to represent multi-component datasets by projecting onto relatively simple diagrams on two-dimensional surfaces. These projections are used to represent a variety of phenomena ranging from spatial relations to physico-chemical processes. By using projections, it is possible to create simple diagrams as models or analogs of complex and heterogeneous natural systems using a limited number of welldefined “end-member” or stylized variables. Although projections are widely used in professional practice, the construction, use and interpretation of these diagrams often presents formidable barriers to student learning. This is largely due to the fact that diagrams that display projected data are the composite product of underlying scientific and mathematical principles, spatial relations on the diagrams may serve as proxies for physical or chemical properties or processes (thus co-mingling spatial reasoning with conceptual reasoning), there are myriad hidden or understood assumptions in the creation of the projections, and projections seek to decrease the “dimensionality” (or degrees of freedom) of multi-component (or multi-variable) systems. Additional layers of information may be superposed on projected diagrams by contouring data, using color or other symbols to distinguish discrete populations of data, imposing gradients of related variables (e.g. isotherms on composition diagrams), or using multiple projections to demonstrate time sequences that elucidate processes (e.g. before/after relations; certain types of animations). Thus, the simple forms of graphical projections may belie numerous layers of information that attempt to explain complex and sophisticated relationships in nature. In striving for simplicity in presentation, diagrams that present projected data may confound student understanding due to lack of knowledge about the inherent complexities in their development. Recall Plato’s Myth of the Cave (Republic, Book 7): the shadow on the wall is at least one step (and probably more) removed from reality. So, too, are the abstract projections used to represent data of all kinds throughout the geosciences.