# Posts Tagged Conceptual vs formal

## Maximum Overhang – annotated

The article “Maximum Overhang” by Mike Paterson, Yuval Peres, Mikkel Thorup, Peter Winkler, and Uri Zwick won the 2011 David P. Robbins Prize, an MAA Writing Award. This pdf of the article is annotated to point out to students how to write a mathematics paper. The annotations address the structure and content of an introduction, how to integrate equations, text, and figures, how to guide the audience through the content, how to cite, etc. The article addresses the question of how far a stack of blocks can extend from the edge of a table. It was published in the American

## Sample proof of correctness

This (fabricated) draft student paper is designed to start a class discussion about when conceptual explanations are needed in mathematical writing. The paper is about an algorithm for finding square roots. The first proof shows that the algorithm is correct, but the point of the second proof is never clearly stated (it shows that the algorithm is efficient). Written by Joel Lewis for M.I.T.’s communication-intensive offering of Real Analysis, based on Rudin’s Exercise 16 in Chapter 3.

## Sample proof for structure discussion

This intentionally mediocre presentation of the proof that convergent implies Cauchy is used to begin a class discussion of when the motivation for a proof should be given before the proof and when it should be given after the proof. Written by Joel Lewis.

## Order of quantifiers worksheet

This worksheet is about how changing the order of quantifiers changes the meaning of a mathematical statement. It was created by Todd Kemp and modified by Kyle Ormsby for M.I.T.’s communication-intensive offering of Real Analysis.

## Notation and LaTeX table & fig HW

The logic exercises in this assignment require students to translate between formal notation and conceptual language, to learn to LaTeX a table, and to include a figure in a LaTeX document. The assignment is from the second week of M.I.T.’s communication-intensive offering of Real Analysis. It was developed by the 18.100C team, especially Todd Kemp and Joel Lewis.

## Definition writing HW

In this assignment from M.I.T.’s communication-intensive offering of Real Analysis, students develop and evaluate various definitions for the notion of a “gap” in a set. The assignment was developed by the 18.100C team, especially Craig Desjardins and Joel Lewis, with modifications by Kyle Ormsby and Susan Ruff. This is the first assignment of the term that requires students to use LaTeX, so students must submit at least one LaTeXed page two days before the assignment is due. This “draft” due date ensures that they devote time to figuring out the basics of LaTeX early enough that they can devote time

## 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. Recitation The following topics were addressed in class discussions: Should conceptual explanations &/or examples be given before or after formal statements? Discussion

## Balancing conceptual with formal

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