**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 countability and metric spaces (Rudin pp. 24-35). Likely trouble spots for students at this point in the term are negation, multiple quantifiers, abstractness of metric spaces, and “higher infinity.”

**Authors: **The recitation was developed primarily by Todd Kemp.

**Communication objectives:** Translating among mathematical concepts, mathematical language, and notation. Making tables and including figures in LaTeX.

**Recitation & Assignment:** Most of the recitation is devoted to working in small groups on a worksheet on order of quantifiers. Other brief topics prepare students for the assignment: reviewing truth tables and briefly giving tips for structuring LaTeX code for tables so it is easy to read and edit. Assignment: LaTeX Exercise 1 (original) or LaTeX Exercise 1 (current).

In Fall ’09, order of quantifiers was challenging enough for students that it was revisited later in the term.

In some semesters, this recitation has been combined with the recitation on proof by contradiction; however, including both activities in one recitation is *very* rushed, and students later commented that they would like more time to be devoted to understanding quantifiers.

### Instructor observations

Kyle Ormsby writes about the worksheet on order of quantifiers, “Students were surprisingly sharp, though I worry that students with less experience may have withheld concerns given their peers’ aptitude with the material. Students who finished quickly got to ponder why sin(pi.n/100) does not take the value -0.1; in retrospect, this may not be the best brain teaser at this point in the course (can you approach the problem without a knowledge of algebraic integers?) and I recommend changing that sequence to something similarly challenging but more approachable given the technology the students have at hand.”

Mohammed Abouzaid writes, “The worksheet for the second week about quantifiers should be considerably simplified. There are about 15-20 minutes of content that should not be rushed through at the beginning of class (including making general remarks about LaTeX if some of them had difficulties), leaving little time for them to discuss the worksheet, as it stands, in groups. If I were to teach this in the future, I would probably have only 4 or 5 different properties, and make sure that all the examples of sequences that we give them are completely elementary (I’m thinking only things built from adding constant sequences, the sequence 1/n, and the sequence n). They have plenty of practice, in class, thinking about harder sequences.”

Mohammed Abouzaid writes, “Second writing assignments: Most students failed to go beyond attempting a word by word translation of the mathematical symbols in the second problem. This should be addressed by (1) ensuring that the examples they produce in their small groups during the class exercise are polished beyond this stage and (2) reminding them of the analogy of translating from a foreign language. In particular, they should be encouraged to break up their rewriting of a given logical expression into sentences connected by appropriate transition words (as opposed to one long sentence).”

Peter Speh writes, “Here are some comments on the second homework assignment ordered by question number:

1) The formula is too long to fit if students write out earlier columns of the truth table. I don’t think this is a problem, but students should probably be warned and instructors should be aware of it.

2a) There is ambiguity as to what is meant inside the last set of parentheses. This could be sorted out through an additional set of parentheses. Most students answered that this was equivalent to A having a least upper bound of s. One student gave the answer that the statement is equivalent to the set A having at most one element, which is correct given an alternate grouping of the last three quantifiers.

2b) A lot of students mistook the statement to mean sup(A) is a limit point of A, which would require a < sup A in the statement. If doing this again, I would change the given statement to have “a < sup A” instead of “a <= sup A”. As it is, the statement holds for any non-empty totally ordered set, which makes it difficult for students to give a good answer.