Professors Steven Jones, Doug Corey and Dawn Teuscher recent had a paper titled “Conventions and Context: Graphing Related Objects Onto the Same Set of Axes” published in the conference proceedings for the PMENA (Psychology of Mathematics Education – North America) conference. They have answered a few questions about this paper below:
What is the big problem you hoped to address with this paper?
There are many “conventions” we use within mathematics, but sometimes these conventions are at odds with reasoning about contexts and quantities. The specific convention we examined in this paper was placing related graphical objects (e.g., graphs, vectors) on the same set of axes. This practice can be problematic for contextualized situations, unless the practice is properly understood.
What are some of the key ideas in the article?
In this paper, we first described the convention using the examples: graphs of functions/inverse functions, graphs of functions/derivatives, and input/output vectors for matrix transformations. We then demonstrated that if the functions/vectors are contextualized with real world quantities (e.g., a function of temperature over time or a vector representing points and rebounds), placing the graphical pairs on the same axes conflicts with the real world meanings, because the axes have to simultaneously refer to different quantities. We then provide the outline of a teaching trajectory meant to permit quantitative reasoning about the objects on separate axes, and then gradually develop the convention in such a way that students see it explicitly as a convention and understand its benefits and limitations.
What are some of the main ideas you hope your audience will take from the article?
Sometimes in mathematics we “do” things without thinking carefully about why we do them or what kinds of reasoning are involved in doing them. This paper hopefully helps teachers see one common convention and possible issues it creates if not properly understood as a convention.
Abstract:
Several researchers have promoted reimagining functions and graphs more quantitatively. One part of this research has examined graphing “conventions” that can at times conflict with quantitative reasoning about graphs. In this theoretical paper, we build on this work by considering a widespread convention in mathematics teaching: putting related, derived graphical objects (e.g., the graphs of a function and its inverse or the graphs of a function and its derivative) on the same set of axes. We show problems that arise from this convention in different mathematical content areas when considering contextualized functions and graphs. We discuss teaching implications about introducing such related graphical objects through context on separate axes, and eventually building the convention of placing them on the same axis in a way that this convention and its purposes become more transparent to students