If one of your objectives is for students to learn to communicate as mathematicians, then consider creating a presentation context that is comparable to those encountered by mathematicians (e.g., conference talk, colloquium talk, poster presentation, lecture to students, informal presentation to collaborators).

### Short Talks

Assigning short talks at the start of the term

- acts as an ice breaker and helps the class get to know each other
- gives you a quick way to assess which students will need the most help with presenting
- helps the students to start becoming more comfortable speaking in front of the class

If students will do a large project as part of the course, a short talk could be an introduction to what they plan to work on.

### Chalk Talks

Students may assume that chalk talks require little preparation; however, many students find that juggling speaking and writing is much more challenging than they anticipate. Consider recommending or requiring the following forms of preparation:

- Schedule a rehearsal enough in advance for the student to recover once it becomes clear just how challenging presenting will be.
- Advise students to plan exactly what they will write on each board. They should review these “board sketches” to ensure that the boards will make sense to anyone who stops listening for a moment to think.
- The few students who have particular trouble focusing their speaking may benefit from the exercise of writing out and revising what they plan to say, but they should not memorize this script or use it during the presentation.

### Slide Talks

In many branches of mathematics (and science in general), it is now expected that talks be given using computer slides projected on a screen. Although preparing the slides requires students to think through the talk ahead of time, students often make the mistakes of including dense mathematics on the slides and rushing through the slides when they present. Consider reviewing a draft of the slides. Again, a rehearsal is very helpful.

### Choosing suitable content

There are various issues to consider depending on whether you assign the material to present or students choose their own presentation topics.

#### Assigned Topic

If you assign students the material, consider the following questions:

- Is the source material at a suitable level? How much mathematical maturity is needed to absorb (and then teach) the material presented? Is the exposition in the source material clear, correct, and complete?
- Is the amount of material assigned reasonable given the time constraints on the presentation? An important presentation skill is deciding what to present: how much flexibility are you willing to give the students about omitting or restructuring the material? If you are open to them deciding to omit certain portions, make sure they know.
- How is the material related to what has been discussed in class already? It may be helpful to speak with the student about what they might need to review and what they can assume the audience knows. If the material foreshadows future material, let the student know so s/he may give the audience this helpful information.

#### Student Choice of Topic

If students choose their own material, many of the issues for assigned topics are still relevant. However, the added complication is that the students may not be able to answer these questions themselves when choosing their material. It may be helpful to require consultation on the subject matter with you well ahead of the presentation.

#### Talks on Final Project

A capstone project (final paper) in the course is a good opportunity for a final student presentation. Students often enjoy the opportunity to speak on material they have worked on for a while. If most presentations up to this point in your course have been chalk talks, the final presentation is a good time to introduce the students to slides and to talk about how changing the “props” of a talk impacts the style of presentation. The transition from giving chalk talks to giving a slide talk is challenging, so a practice presentation is essential.

### Should students present alone or with a collaborator?

It’s common for each student to present alone, but an advantage to having students present in pairs or teams is that the students can help each other to present well. Below are some observations from educators who have assigned pair and team presentations.

#### Pair presentations in M.I.T.’s *Seminar in Number Theory*

Scott Carnahan experimented with having students present math in pairs. He assigned a general topic for each presentation, and the students covered some amount of material that seemed reasonable for a 30 minute talk, coming up with their own strategies for presentation.

Scott writes, “In almost all of the pair presentations, the students decided to split the talk roughly in two, and each student talked for about 15 minutes. In some of the cases, one of the students in the pair had a weaker background or less confidence, and just gave some background information while the more confident student presented the “meat” of the talk. I don’t know how much time they spent preparing, so I can’t tell if this is a just division of labor.

“There were a few attempts at more innovative presentation styles. For one pair, they allocated half of the board space to each student, and took turns talking, where one student introduced the general theory, and the other student wrote examples. It wasn’t an exceptionally polished performance (they had some synchronization problems), but it showed some potential.

“I should mention that one key advantage of pair presentations is that it forced the students to work with each other for preparation, instead of only spending time with me. For the solo presentations, I had been telling them what I personally liked to see in a presentation, but I think hearing another student’s opinion is valuable.”

#### Teams of 3 in M.I.T.’s *Project Lab in Mathematics*

Susan Ruff writes, “In Project Lab students present in teams of 3. Here are some of the challenges they face:

- If the entire presentation is 45 minutes, each student presents for only 15 minutes. It’s very difficult for students to judge an appropriate amount of material to try to include in 15 minutes. If the timing isn’t flexible, doing a timed practice presentation is essential–usually more than one practice presentation is needed. We schedule a practice presentation with the team mentor and me for a few days before the in-class presentation; then students are encouraged to do at least one more practice presentation on their own.

“Here are a few superficial issues the students often face:

- If the later presenter needs to refer to content presented by an earlier presenter, the presenters need to ensure that the necessary earlier material isn’t erased.
- When one presenter makes a mistake, the others typically either sit silently and cringe or try to whisper the correction. You can point out to the presenters that they should help each other and that they can interrupt professionally: “Excuse me, …” “Thank you.”
- Often the presenters aren’t prepared for how to transition between presenters.
- One of the greatest advantages of presenting in teams is that the teammates can help each other to plan a good presentation–they can point out for each other ways to present information more clearly, and they can together come up with good ideas for how to structure the presentation and how to handle the most challenging parts of the content.

“A few years ago one 821 team did a shorter presentation in which the teammates assumed different roles: one was the narrator (the person who provided verbal guiding text or meta explanation) while the others did the mathematical heavy lifting as needed. This dynamic worked remarkably well for the short presentation. For a longer presentation I could imagine that one challenge of this approach might be that the narrator is on stage all of the time, so must remain still to avoid being distracting.”

### Writing a clear assignment

Specify the details of the assignments: content, purpose, timing, format (chalk? slides?), grading, educational objectives. If students don’t yet know each other or if you would like students to target an audience other than each other, also indicate how much background knowledge they may assume.

### Examples

- See the examples above under “Should students present alone or with a collaborator?”

Please contribute additional examples of presentation assignments, along with commentary on the effectiveness of the assignment.