Function
"If undergraduate mathematics does nothing else, it should help students develop function sense"
-- National Research Council, 1988
Functions are centrally important objects of study in mathematics, and yet, before taking an Introduction to Proofs course, most students have experienced functions only as rules for computing. In our Intro to Proofs courses, we introduce functions as objects of study, often by defining them formally as sets of ordered pairs, (x, y). So, there is a disconnect between students' past experiences with functions as computational processes and their structural definition in proofs courses (Sfard, 1991). This disconnect can lead instructors and students to experience related EOs in the classroom, especially as students are expected to consider properties of functions, their inverses, and their compositions.
Identified Epistemological Obstacles
Just as holding logical implications as objects can empower students' abilities to transform and quantify implications, holding functions as objects should empower students to operate with them. However, constructing functions as objects may be just as challenging as it is powerful.
In the following video, the instructor is leading the class in a discussion about the validity of statement #3 (see image below). Notice in the video that students seem to be conflicted about whether f is a fixed object or simply an input-output machine of any kind whatsoever. The latter conception fits with an action conception of functions. To hold a given function as an invariant relationship between variables requires more.
Students began to address related challenges in groups. The following video shows the Triangle group grappling with the three statements shown in the image above. Notice that they rely heavily on the graphs and graphical properties (e.g., whether the graph has a hole in it) when considering possible examples or counterexamples for each statement.
Transformations of functions may include adding functions, composing functions, and inverting functions. Such transformations and conceptualizing their results as functions may be especially problematic for students who treat functions as actions (as an input-output machine or a rule for computation).
The instructor supported students understantings of transformations of functions by presenting functions as mappings from a domain to an image. The inverse of a function, then, would map from this image to some pre-image in the domain. In the video shown below, we see the Circle group grappling with this conception as they attempt to identify the inverse function for f(x) = x - 1. Note that the students begin with a simple algebraic manipulation, but then discussion turns to the f-1(x) notation for an inverse and how that symbolizes a mapping.
Key properties of functions include whether they are one-to-one, onto, and invertible.
In addition to the properties mentioned above, the instructor (in the video shown below) introduces the necessary property that a function be well-defined. Students in the class begin relating this and other properties to each other. For example, one student wonders whether the well-defined property is the same as the one-to-one property.
After introducing the formal definition for the inverse of a function, the instructor asks students to work in groups on two related questions (see image below)? In the following video, the Hexagon group relates the property of invertibility with the one-to-one property. Notice how students rely on specific examples and the schematic diagram (with images and pre-images) to come to agreement that an invertible function must be one-to-one.
Later in the semester, the instructor challenged students to consider properties of functions in relation to their composition (see image below). Students worked in groups to form and justify conjectures about these relationships. For example, in the following video the Triangle group is considering what they might infer about f and g, given that g of f is one-to-one. Notice that the students rely on the definition of this property, along with the schematic diagram to reason with the given property of g of f and hypothetical properties of f and g.
