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.

Conceptual vs. Formal

Context: This lesson plan is from a weekly communication recitation that accompanies M.I.T.’s Real Analysis (18.100C). This week students learn about completeness and sequences and series (Rudin pp. 42-43, 47-69, 71-75). Trouble spots for students at this point in the term may include properties of continuous functions.

Authors: This recitation was developed by Craig Desjardins and Joel B. Lewis based on a suggestion by Susan Ruff.

Communication objectives: Choosing when to write conceptually and when to write formally.


The following topics were addressed in class discussions:

  • Should conceptual explanations &/or examples be given before or after formal statements? Discussion of when and how to build intuition and when and how to present math formally was sparked by these sample proofs:
  • When this recitation is given later in the term, the following examples from Rudin are also useful:
    • Rudin pp.122-132 on the Riemann-Stieltjes integral sparks discussion of whether to include conceptual explanations before or after formal treatment. See Definition 6.2 and Remark 6.18.
    • Rudin’s proof of Theorem 7.18 generates discussion of the merits of purely formal/technical presentations vs. presentations that include conceptual ideas.

All page numbers are from Rudin, Walter. Principles of Mathematical Analysis. 3rd ed. New York, NY: McGraw-Hill, Inc., 1976.

The end of the recitation focused on figures and LaTeX.


Revise the proof-writing assignment from the fourth recitation.

Instructor comments

The discussion of when to provide conceptual explanations and when to present concepts formally is often folded into the recitation on structuring proofs.

The focus on figures and LaTeX has been folded into the recitation on advanced LaTeX topics.

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