SLO #5  [Definitions] Understand the role of definitions in mathematical discourse:

1. Understand the importance of precise definitions for geometric objects and that necessarily some geometric terms and relationships must remain undefined.
2. Recognize that there are a variety of acceptable definitions for some geometric objects.

The role of definitions in mathematics is a rich area for students’ exploration that is often overlooked. In many math classes, definitions are given by the textbook or teacher. However, in a geometry class, definitions can be a fruitful area for students to explore. Students can propose their own definitions for elementary concepts, such as a square, a triangle, circle, or even a straight line. They can engage in class discussions about verifiable mathematical definitions versus vague descriptive definitions, and they can compare and contrast definitions with different properties included.  For example, when asked to define what a rectangle is, one student might say it is a quadrilateral with four equal angles; another might say it is a quadrilateral that has at least three right angles and doesn’t have four equal sides; another might say it is a quadrilateral with reflection symmetry across the perpendicular bisectors of its sides; yet another might say it is a quadrilateral with four congruent angles and two pairs of congruent parallel sides.  

Classes can have rich discussions regarding both the equivalence and the quality of proposed definitions. Criteria for the quality of definitions could include: (1) use of commonly understood words or previously defined terms, (2) accurately describing what is being defined, and (3) including no superfluous information. One strategy to convey the need for (1) is to “define” two “nonsense” words with definitions that refer to one another and hence have no meaning. Some definitions must involve undefined terms, to avoid infinite regress. A strategy to convey this is to ask students to come up with a definition of a familiar object and prompt them to define the terms they use in their definition.

Choices for definitions necessarily set the context for proving activity. For example, proving that two lines are parallel because they do not intersect can be very different from proving that they are parallel because they are everywhere equidistant. GeT students can also consider how changes in the assumptions within a geometric definition can lead to changes in the interpretations of other terms involving that definition. For example, a circle is often defined as the set of all points in a plane that are equidistant to a given point. If a geometry adopted the Euclidean metric for distance, then the property that distinct circles have a finite number of intersections holds; however, this property is not maintained with the Taxicab metric (Krause, 1975). 

The logical consequences of statements involving a definition include the assumed meanings for terms within a definition as well as the axioms of the system . For those who include significant non-Euclidean topics  there is opportunity to investigate the same definitions using different models. For example, there are no quadrilaterals with four right angles on the surface of the sphere or on the hyperbolic plane, but there are still quadrilaterals with reflective symmetry over the perpendicular bisectors of their sides. On the hyperbolic plane, there are lines that do not intersect but are not everywhere equidistant. On the sphere, there are not any lines that do not intersect, but there are still lines that make equal corresponding angles with a transversal. Some instructors have been surprised to discover that when students spend time in class exploring definitions and then are given a new space to explore on their own, they can productively spend weeks exploring the implications of potential definitions. For instance, what is a circle on the surface of a cone? If a circle is defined as the set of all points obtained by going a fixed distance from a given center in all directions, we get different circles than if a circle is defined as a figure with constant curvature, which is in turn different than if a circle is defined as a closed figure such that every straight line segment from the center to the boundary is the same length. Each of these types of circles have different properties that students can explore.

The taxonomy of geometric objects is closely tied to definitions, and the exercise of classifying objects helps GeT students attend to the ramifications of adopting different definitions and become prepared to support prospective students’ reasoning at different Van Hiele levels (see, e.g., Burger & Shaughnessy, 1986). In elementary school, students are taught how to identify and classify different quadrilaterals as rectangles, rhombi, squares, or none of the above. As definitions become formalized in middle and high school geometry, they become associated with increasingly generic representations. It is in the GeT course that students consider the results of adopting alternative definitions for geometric terms. For example, trapezoid is typically defined inclusively in college geometry courses (a trapezoid is a quadrilateral with at least one pair of parallel sides) but it is sometimes defined exclusively in elementary and secondary courses (a quadrilateral with exactly one pair of parallel sides). Other terms commonly encountered in secondary geometry for which GeT students could discuss the consequences of adopting definitions supporting exclusive or inclusive meanings include: whether coincident lines are types of parallel lines, whether kites are types of rhombi, and whether the identity transformation is considered to be a type of rotation or a type of translation. Determining whether and when definitions have equivalent meanings and the consequences of adopting exclusive or inclusive definitions prepares GeT students for the varieties of geometric definitions they may encounter in teaching secondary geometry. 

References

Burger, W. F., & Shaughnessy, J. M. (1986). Characterizing the van Hiele levels of development in geometry. Journal for research in mathematics education, 17(1), 31-48.

Krause, E. F. (1975). Taxicab Geometry. Addison-Wesley Publishing. Menlo Park, CA.