Guiding the audience through the content

Guiding text gives signposts to the reader: how is the paper or presentation structured? what strategy does this proof use? what is this paragraph about?

The following examples illustrate how guiding text can be used to

communicate the structure of a paragraph or a logical argument

• “We will prove that x² = 2 by ruling out the other two possibilities: x² < 2 and x² > 2…”
  –R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis 3rd ed., p. 41.
• “…Now assume that x² > 2. We will show that it is then possible to find m ∈N such that x – 1/m is also an upper bound of S, contradicting the fact that x = sup S…”
  –R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis 3rd ed., p. 42.
• “…We want to prove that m is a rotation about some point. It is clear that an orientation-preserving motion which fixes a point p on the plane must be a rotation about p. So we must show that every orientation-preserving motion m which is not a translation fixes some point…”
  –M. Artin Algebra Prentice Hall 1991, p.159.

communicate the structure of a section

• In this section, we try to use Kolmogorov complexity to solve the problem of defining random infinite sequences. A natural attempt to do this would be to declare an infinite binary sequence ω random if all its initial segments are incompressible; that is, if there is a constant c such that for all n, the n-length prefix ω_1:n has C (ω_1:n) ≥ n – c. We will prove (as a consequence of Theorem 2.5.1) that such sequences do not exist. To obtain a nonempty class of random infinite binary sequences, we will weaken the restriction on their initial segment complexity. In particular, it will turn out (Theorem 2.5.4) that all random sequences (in this sense) satisfy that C (ω_1:n) ≥ n – log n – 2 log log n for all n. However, Corollary 2.5.1 illustrates that the Kolmogorov complexity of initial segments of random sequences also dips unboundedly low infinitely often. Thus, the Kolmogorov complexity of random infinite sequences has unbounded complexity oscillations.
  –from Mia Minnes’ Paper Workshop Handout, based on a paragraph from Section 2.5.1 of Li and Vitanyi’s An Introduction to Kolmogorov Complexity and Its Applications.
• Theorem 3.1. 3-colorability is a knot invariant.
  Our approach is slightly indirect, and involves the arithmetic of the field F₃ of integers modulo 3…We will show first the following:
  Lemma 3.2. A vector v ∈ (F₃)ⁿ is a coloring of P if and only if Av= 0.
  …As an easy consequence, we get:
  Lemma 3.3. The knot projection P is colorable if and only if rank(A) < n – 1.
  After that, we examine how the matrix changes under Reidemeister moves:
  Lemma 3.4. The effect of any Reidemeister move on the matrix A can be written as composition of the following operations, and their inverses…
  From linear algebra, we know that the rank of a matrix is not affected by such operations. Hence, Theorem 3.1 follows directly from Lemmas 3.3 and 3.4. The rest of this section contains the proof of the Lemmas.
  –The example above is from Paul Seidel’s slides Writing a Project Report.

communicate the structure of the paper

• “Our project studies knot 3-colorings…It will turn out that…the number of 3-colorings, is a knot invariant. This is the main result of the paper (Theorem 3.1), and will be shown using a case-by-case analysis of knot deformations.”
• “Knot colorability can be generalized from 3 colors to n, for any n ≥ 3. We discuss this generalization in Section 5.”

These examples are from Paul Seidel’s slides Writing a Project Report..

Various handouts illustrate the use of guiding text:

• An annotated journal article, “Maximum Overhang” by Paterson et al, American Mathematical Monthly 116, December 2009. See page 773 (page 11 of the pdf).
• An annotated proof using the pigeonhole principle, from lecture notes for M.I.T.’s Principles of Applied Mathematics.
• An annotated proof from an Analysis textbook
• An annotated student paper from M.I.T.’s Undergraduate Journal of Mathematics: “Finding Generators for Invariant Rings” (annotated)
• An effective strategy is to take a published paper of your own and mark your own guiding text (and other writing strategies) for your students.

Using guiding text in presentations

In presentations, guiding text is useful not only in proofs (“First we will show…then we will show…) but also at transitions within the presentation. (“We have just…. Are there any questions? Now we will…) A useful trick is to give an outline early in the presentation and then return to it throughout the presentation to keep track of progress and how everything fits together.