A kind communicator can help the audience to build conceptual understanding of a new idea, for example, by including conceptual explanations, well-chosen examples, and/or figures. There is always a danger, though, that the audience will misinterpret conceptual explanations or draw inappropriate conclusions from the specifics of figures or examples. To help the audience understand the concept correctly, important concepts should also be presented formally.

Students often have a difficult time appropriately deciding when to build conceptual understanding and when to present ideas formally; additionally, students often don’t realize that conceptual explanations must be as carefully constructed as formal presentations.

These challenges can be addressed by commenting on student writing and presentations individually as needed. Here are a few examples of how these challenges have been addressed with a class as a whole:

### A recitation and writing exercise

One of M.I.T.’s offerings of *Real Analysis* is accompanied by a weekly communication recitation. In the first recitation of the term, students are presented with a new concept (the notion of gaps in sets). In class they suggest and discuss different ways to define the term “gap.” During class discussion, the instructor points out advantages and disadvantages of different explanations that students offer. The assignment is then to write up a few different possible definitions, providing both the intuition behind the definition and the rigorous/formal definition itself. Students receive individual feedback on their writeups.

The lesson plan and assignment are here.

Later in the course, further recitations address when to give conceptual explanations and when to present ideas formally:

Recitation 6: When to write conceptually vs formally

Recitation 10: Communicating to different audiences

### Sample presentations

Sample presentations have been designed to spark class discussion about the interplay between formal and conceptual. See the page “Sample presentations: examples and cautions.”

### Sample proofs

The following sample proofs are designed to spark discussion about when to draw attention to the key idea of a proof: before, during, or after the formal presentation?

- Sample proof that convergent implies Cauchy
- Sample paper proving the correctness of an algorithm for computing square roots.
- This annotated article, “Maximum Overhang,” illustrates how a conceptual overview can be helpful at the beginning of a long section (see p. 769, which is page 7 of the pdf). This article also illustrates how the structure of a complex argument can be communicated by presenting the various Lemmas, Propositions, and Theorems first to indicate how they are related to each other, and proving the individual statements later. (See page 772, which is page 10 of the pdf.)

This webpage needs resources for how to build conceptual understanding (e.g., choosing effective examples and counter-examples). Please feel free to suggest or contribute resources.