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.

# Information Order and Connectivity

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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 closed sets and compact spaces (Rudin pp. 34-38). Likely trouble spots for students at this point in the term include equivalent forms of compactness and how to prove TFAE theorems (“The following are equivalent:…”).

Authors: The recitation was developed by Susan Ruff based on the article “The Science of Scientific Writing” by Gopen and Swan; the sample paragraphs were written by Joel B. Lewis.

Communication objectives: Ordering information so explanations flow logically.

### Recitation

Written on the board as students enter:
Read the writing samples # and % in some random order. Decide which you prefer and why.

At the start of class, I take a show-of-hands vote. Version # has always been preferred by most, but not all, of the students. One term we did a quick on-the-spot statistical test to determine whether reading order was affecting the results: the preference for # was obviously robust to reading order.

I ask students to briefly explain why they chose the writing samples they did–this gives me an immediate sense of how the students think about writing. At the end of this discussion I point out that readers vary and it’s not possible to give writing rules that will enable you to achieve your desired effect with all readers, but there are guidelines you can follow to maximize the chances that readers will interpret your text in the way in which you intend.

I then give a mini-lecture on information order and connectivity.

In short, each sentence should begin with information that is familiar to the reader. Ideally, this information should have been fairly recently discussed in the text. This familiar information can then be used to introduce the sentence’s important new information, which appears later in the sentence.

A desirable side effect of ordering information within sentences in this way is that each sentence is likely to connect tightly to the preceding sentence and to the following sentence, thus creating a flow (or “connectivity”) that makes the text easy to read. (e.g. “I need calcium. Cheese has calcium. I’ll eat cheese.” vs “I need calcium. There’s calcium in cheese. I’ll eat cheese.” The second version has better information order and connectivity and, for most readers, is easier to read. The effect is minor when the content is simple, but becomes more pronounced as the content becomes more challenging.)

In the following shorthand, A–>B is a sentence in which familiar information A is used to introduce important new information B, while C<–B is a sentence in which new information C is explained by familiar information B, which appears later in the sentence.

Easy for readers: Old–>New.
Easy for readers: A–>B. B–>C. C–>D.
Hard for readers: A–>B. C<–B. C–>D.
Hard for readers: A–>B. C–>D.

Students test their understanding of these ideas by doing this in-class worksheet, which progresses from simple exercises to more complex. A few notes about this worksheet:

• The first exercise is about the verb: the sentence’s verb should communicate the action of the sentence. Although engineers often hide the action of the sentence, mathematicians rarely do, so I often skip this exercise.
• The definition exercise is intended to spark discussion: mathematicians have a convention of stating a term (important new information) before defining it (connecting it to familiar information). This order of information is backwards, but in this case it’s probably more important to follow mathematical conventions than to follow the information-order guideline. Both can be satisfied by informally introducing a term before defining it (e.g., “We can build a heap by repeatedly fixing headless heaps. A headless heap is…”)
• In a 50-minute recitation, we rarely have time to get to question 5.

Once students understand the theory of information order and connectivity, we return to the writing samples from the beginning of the class. Students identify problems with information order in % and how those are fixed in #.

I conclude the class by showing students how they can draw a diagram of the logic in a proof or paragraph. The third paragraphs of # and % provide a good example. Once the logic is sketched, we can see that the logic is in some sense two-dimensional because it has branches. As a result, the logic is difficult to communicate in a single connected one-dimensional line of text. By looking at the diagram, we can see how the two versions of the paragraph handle this challenge.

### Assignment

The first few times this recitation was offered, the assignment was to critique a classmate’s writing. The rationale for this assignment was that it’s easier to identify problems with connectivity in another’s writing than in your own writing. (If your eyes glaze over or your mind wanders as you read, the chances are good that there are problems with connectivity in the text: you can find them by analyzing information order.)

Although some students commented that doing peer critique helped them to solidify their understanding of information order & connectivity, a drawbacks of this assignment was that the peer critiques were supposed to address any relevant issues, so there was no need for students to consider information order and connectivity in particular. Students reported at the end of the term that they didn’t feel able in their own writing to order information to create connectivity.

In Fall 2011 we tried a new approach: each student received an individualized assignment target to information order and connectivity. I read each student’s first writing assignment and identified two paragraphs for the student to revise, one on which I noted issues with information order and connectivity, and one on which I provided no comments. The purpose was for students to use the first paragraph to practice fixing problems with information order and connectivity and to use the second paragraph to practice finding (as well as fixing) such problems. For students whose writing already exhibited good information order and connectivity, I either assigned problem 5 from the connectivity worksheet if we hadn’t gotten to it in class, or I told the student to find writing related to one of their math classes that had problems with information order and connectivity, analyze it, and revise.

### Instructor Observations

In the future I think each student should be assigned the same carefully-constructed paragraph to revise for information order and connectivity. Because this assignment is short, it could be combined with a peer critique assignment.

Preparing individualized assignments for the students was very time consuming, and it was rare that I was able to find two suitable paragraphs for the student to revise. Furthermore, the one advanced student who was asked to find a sample elsewhere to revise chose one that exhibited many confounding problems with both the writing and the mathematical content, so she required assistance to complete the assignment. In short, this assignment is likely to be most effective if it is carefully constructed rather than “found.”

The writing samples are currently about the topic of the first assignment (gaps in sets). It might be helpful to students if the samples were instead about likely trouble spots at this point in the term (e.g., equivalent forms of compactness).

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### What is Math Comm

MAA Mathematical Communication (mathcomm.org) is a developing collection of resources for engaging students in writing and speaking about mathematics. The site originated in the MIT Department of Mathematics and was expanded through support from an NSF grant.

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