Analyzing Definitions: A Quick & Easy Class Activity

Analyzing, comparing, and using precise definitions is one of the most important learning goals of my Real Analysis class. Here’s one activity from early on in the course that helped students come closer to achieving this goal.

Pre-Class Reading

Prior to our second class, students read Section 2.1 of our course textbook, A Radical Approach to Real Analysis. In it, Bressoud presents the following definition for the Archimedean understanding of infinite series:

The value of an infinite series, if it exists, is that number T such that given any L < T and any M >T, all the finite sums from some point on will be strictly contained in the interval between L and M. More precisely, given L < T < M, there is an integer n, whose value depends on the choice of L and M, such that every partial sum with at least n terms lies inside the interval (L, M).

Warm-Up

At the beginning of each class, students complete a short quiz, which I call a Warm-Up. On the second day of the course, the first question read:

Explain what is meant by the Archimedean understanding of the value of an infinite series.

When students were engaged in another activity, later in class, I quickly compiled their definitions into one document. Three of the student definitions are below.

The value of an infinite sum is the target value T. If T is the sum of the infinite series then it is possible to show that for all L<T and all M>T there is a point in the summation that all remaining finite sums will be greater than L and less than T.
To find the target value T which we are certain the sum approaches we bind it by finding values L and M such that L < T < M and after a certain term all values of the sum fall between these values.
The value T of a series exists if and only if we can define an arbitrary interval (L, M) that bounds T for which after a finite number, n, of terms, will thereafter contain the partial sums.

Analyzing Definitions

Later in the class, I passed around the document containing each student’s response to the first Warm-Up question. I assigned two definitions to each pair of students, and asked them to critique the definitions for both correctness and form.

The discussion that followed was excellent: students were able to identify all of the errors in their definitions and had suggestions for improving each definition. They compared how serious different errors were, and discussed the consequences of changing various parts of the definition.

Learning Benefits

  • Students received immediate and comprehensive feedback on their work. By the end of the class, they understood what they got right and what they got wrong, and were more careful with their wording on the next day’s warm-up.
  • Seeing multiple similar definitions side-by-side allowed students to identify subtle differences.
  • The fact that these definitions came from themselves and their classmates made the activity authentic. Critiquing definitions wasn’t some imaginary ‘pretend you’re the teacher’ exercise; they were helping to improve the understanding.
  • Students began to appreciate the need for precision, which is one of the driving themes for this course.
  • It was easy to implement and didn’t take very much class time (only about 20 minutes total).