Using 'Objects' to Object to Objectification

My high school students and I spent a lot of time exploring functions in our pre-calculus and algebra classes. Indeed, the function concept has been described as the biggest single vehicle around which essential high school mathematics can be organized. The basic level of the function concept is "function as a process." The National Council of Teachers of Mathematics requires that students view a function as an object that can itself be acted on. Another useful way of viewing a function as an object is by subjecting it to being classified in various ways.

Because our high school had a strong mission of pluralism and tolerance, I was moved to see this traditional mathematics content as an opportunity to be enhanced with a metaphor for tolerance — a goal not commonly discussed in mathematics classrooms.

First, I gave my students a table where each row had a different function. Each column had a trait by which the function could be classified as having (or not), such as whether the function was even, odd, increasing, decreasing, continuous, 1-to-1, going through the origin, or satisfied f(a+b) =?f(a)+f(b).

After filling in the table with "yes" or "no," students noted how difficult it is to find a single simple property shared by all, or to find a single row that is uniquely defined by any one of its traits. And yet doesn't most intolerance stem from assumptions in the form of "all people in Group Y have trait X"?

As a follow-up discussion or writing exercise, I ask students to explore this problem: "In popular usage, the phrase ‘treating as an object' has negative connotations, because it implies an entity as rich as a human being can be reduced to a single dimension, such as gender, ethnicity, financial status, sexual orientation, religion or occupation. Would it be just as foolish to say that we know everything about a function or its behavior from one particular classification of it? Why?"

This activity can be easily adapted in other math classes by changing the row and column headings in the table. Students in younger grades can be given a version using simple whole numbers (i.e. 1-10) instead of functions. Possible "traits" of numbers include whether a number is even, prime, composite, square, perfect, triangular, Fibonacci or factorial.

Teaching Tolerance Recommends
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