Translating concepts into math language

Context: This lesson plan is from a weekly communication recitation that accompanies M.I.T.’s Real Analysis (18.100C). This plan is for the first recitation of the term. This week in lecture students learn about sets & fields and the real numbers (Rudin pp. 1-17). Likely trouble spots for students at this point in the term are absorbing many new definitions at once and expressing math concepts formally and rigorously.

Authors: The lesson was developed primarily by Craig Desjardins and Joel B. Lewis and has been further refined by Kyle Ormsby. The commentary below is by Joel Lewis and Susan Ruff.

Communication objectives: Translate mathematical concepts into clear mathematical language; write precise definitions; setting text and simple math with LaTeX, including using definition environments.

Recitation

In the first recitation of the semester, we led a discussion of how one should formalize the notion of a “gap” in the rational numbers. (Rudin’s text begins with a proof that the square root of two is not a rational number, before going on to build up from ordered field axioms. The course did not cover the Dedekind cut construction of the reals, Rudin’s Appendix to Chapter 1.) We gave the students a few minutes to discuss the question in small groups (with the instructor walking around and briefly talking to each group) and then brought the discussion to the board.

Students wrote their definitions on the board (simultaneously) and then we discussed them. Doing so provided opportunities to discuss precision of wording, which definitions capture our intuition about the “gappiness” of particular sets, and the characteristics of a “good” definition.

Assignment

We then gave the following assignment for homework:

Discuss possible meanings of the sentence, “There are gaps in the set S,” and discuss how they apply to the set Q of rational numbers. You should give at least one definition that allows you to conclude that there are not gaps in the rationals, and you should give at least one definition that allows you to conclude that there are gaps in the rationals. The order in which you present these definitions does not matter – do what works!

This paper is not intended as a debate piece. Rather, you should try to explain how the informal word “gap” can be given more than one precise definition, and how these different definitions can lead to different answers to the question, “Do the rational numbers have gaps?” Along the way, you may find it fruitful to consider how your definitions apply to other sets (for example, Z, R, (−∞, 0] ∪ (1, ∞), etc.) and to discuss the strengths and weaknesses of your definitions (for example, whether they are extrinsic or intrinsic). You are invited to invent names for the properties you define, in which case you should try to make a good choice of terminology.

The paper should be two to three pages long and written for a peer, such as an undergraduate student at MIT who has not taken and is not taking 18.100C. Your writing should be clear, precise and unambiguous, but shouldn’t be so technical that only someone taking 18.100 could follow it. You are welcome to cite useful definitions or theorems from Rudin, in which case you should include a simple bibliography.

A more tightly specified version of the assignment, by Kyle Ormsby, is available here. When the assignment is introduced at the end of recitation, students are provided with an introduction to LaTeX, a template, and a verbose template. The verbose template should ideally be revised to more accurately represent the sort of mathematics the students will be writing.

In Fall 2009, we also assigned a revision of this paper based on instructor comments.

Instructor Observations

The first-day introduction to the recitations is kept very brief, but we’ve found that it is important to clearly state the communication focus of the recitations: otherwise the students who expected a traditional recitation express great disappointment at the end of the term.

In bullet-point format, here are my [Joel’s] main thoughts about this assignment:

• This is a good assignment to get students writing and thinking about thinking carefully and precisely: it requires essentially no background, requires only minimal LaTeX skills (few complicated symbols), and allows a lot of focus to be placed on communicating precisely in mathematics.
• Doing a revision of the first assignment led to what we felt was over-emphasis on the content of the assignment. Since the question of whether or not the rationals have gaps is not mathematically very interesting, and stretching the discussion of it over three weeks (one for writing, one for grading, one for writing) probably didn’t add much of anything.
• This assignment leads to discussion of a variety of potentially interesting concepts. Topics that arose included the following:
  • “Betweenness” as a property of both ordered sets and geometric objects (convexity)
  • Intrinsic versus extrinsic properties
  • Algebraic versus topological/geometric definitions of “gaps” (and, for example, why the unsolvability of x^2 = 2 in Q is fundamentally different from the unsolvability of x^2 = -1).
  • Ideas related to density and approximation
  • Open and closed sets/intervals
  • Abstract constructions of “pathological” objects. (Okay, I’m stretching here: the point is that students are often surprised by the existence of an object like R ∪ {ε} where 0 < ε < r for every positive real r.)
• Class discussion typically reaches the following two definitions:
  1. A totally ordered set has a gap if there exist two elements of the set with no element of the set between them.
  2. A totally ordered set has a gap if it can be divided into two halves such that every element of one half is smaller than every element of the other, and the small half has no LUB while the large half has no GLB. (Or something similar to this.)
• Some industrious students will go beyond those two definitions, inspired by interesting examples of sets that “obviously” have gaps that aren’t detected by the definitions above. (Examples: Z has gaps not detected by the second definition; (-1, 0) ∪ (0, 1) has a gap not detected by the first definition.) Many students will not.
• Pitfalls include:
  • A number of students like definitions of the form, “A set has a gap if there is some element not in the set between some elements of the set.” This definition has all sorts of problems, and in Spring 2010 I led some extended discussion of the difference between intrinsic and extrinsic definitions and how this definition only makes sense relatively to some superset. This had the unexpected consequence of leading more students to use a definition of this form, simply taking the superset to be the real numbers. If I were assigning this again, I would require the definitions to be intrinsic.
  • Some students will make assertions that suggest they believe that there is no ordered superset of the real numbers, even if an example has been presented.
• I was initially skeptical that this assignment would work well, and my opinion of it improved after we first assigned it. Craig, who is primarily responsible for the idea of the assignment, had his opinion of it fall over time.