Mathematical Communication is a developing collection of resources for engaging students in writing and speaking about mathematics, whether for the purpose of learning mathematics or of learning to communicate as mathematicians.

Audience Awareness

Context: This lesson plan is from a weekly communication recitation that accompanies M.I.T.’s Real Analysis. This week students learn about integrability and the fundamental theorem of calculus (Rudin pp. 128-136). Possible trouble spots for students include multiple quantifiers, formalizing concepts, and uniform continuity vs. convergence.

Authors: This recitation was developed primarily by Joel B. Lewis, Craig Desjardins, and Susan Ruff

Communication objectives: Analyze the rhetorical context of a communication and design the communication appropriately.


Pair or small-group discussion: How would you explain the mean value theorem to a physics major who’s asking for help in a required math class? To a potential employer who has asked you in an interview what you’re doing in math class right now? To a ten-year old relative at Thanksgiving who idolizes you, likes math, and asks what you’re doing in math class?

Students who finish early are asked to sketch a graph showing how much detail to give depending on the background knowledge possessed by the audience.

Class discussion:

We begin by gaining context for the employer scenario by asking the class who they see as potential employers. (F09 answers: finance, Pixar, Google, university; additional answers S10 include surf shop, State Department, Facebook; F10 and S11 were more pessimistic with much uncertain silence and “Starbucks.”).

We then discuss the questions and the graph above.

After some discussion, we turn the focus more explicitly to rhetorical context:
“What aspects of the situation affect how you explain the mean value theorem?”
Point out the following ideas if they don’t come up in the discussion:

  • What does the audience know (or *not* know)?
  • What is the audience’s goal?
  • What’s *your* goal?
  • (What are the constraints of the mode of communication? e.g. phone interview, time constraints, background noise)

Time permitting, we show how rhetorical context can be applied in a wider range of contexts, such as the guy on the bus next to you looking over your shoulder at Rudin and asking what you’re reading.

Additional prompt for discussion:

Summarize the Stieltjes integral; then ask students how they would explain the Stieltjes integral to a prospective employer in finance (use finance example: weight recent values more heavily), to a physics person (dirac delta example), to a number theorist (alpha step function–>integral of f d(alpha) = sum f, so can use tools of integrals for sums).


Choose a statement from real analysis and explain it to three different audiences of your choosing. The three audiences should be substantially different from each other. A recent version if this assignment is here.

Instructor observations

For the graph, some students think at first that those with no knowledge should be given the most detail. We eventually agree that the graph should look roughly like a bell curve: novices and experts should be given the least detail, while math students should be given the most detail.

Be realistic. If the rhetorical context calls for lack of communication or glossing over the truth, then those are the appropriate responses.

I’ve worried that students would consider this recitation to be fluff, but students respond well to this assignment and say that it gives them an appreciation for how much detail to include in different contexts. We’ve discussed whether the recitation should be moved earlier in the term, but students agree that if it were earlier in the term (before a serious tone has been set for the term) it would feel more like fluff.

In the Thanksgiving discussion, we’ve avoided the potential pitfall of making gender assumptions by being specific about the background and interests of the particular grandparent or sibling in question.

Time permitting, it might be helpful to talk about the rhetorical context of the essays required for funding opportunities for undergraduate research, grad school applications, etc. and how that context affects how to write the essay.

As written, the assignment permits students to writing about anything mathematical, but they should be required to write about something from analysis to more tightly tie the recitation to analysis and so peers can critique the writing.

For the assignment, many students choose contexts that naturally call for an oral response rather than a written response, so their written responses can seem somewhat awkward as a result. We’ve discussed the possibility of changing this into an oral assignment. Although students like the idea, our students have ample opportunity to present orally in other classes so the time isn’t justified here. A reasonable middle ground is to give students the option of writing a dialogue. We have not yet tried this in Real Analysis, but it has worked well in other communication-intensive math classes.
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