Having a mathematical idea is only one part of the equation (excuse the pun!); we also need to be able to communicate it. A good example is Fermat writing next to his conjecture that he (purportedly) had a proof that the margin was too small to contain. If Fermat really* did* have a proof, then he could have saved mathematicians 350 years of frustration by going to the effort of communicating it with us!

There are many different dimensions to how we want our students to be able to communicate. Is it in written or oral form? It is to be polished, or in intentionally draft form for revision? Does it present the resolution of mathematical work, or is it work in progress? Who is the intended audience — peers, novices, experts, themselves?

The way we write or talk, and the skills we need for the process, depends on which of these dimensions are at play. I think it is unreasonable to expect that students can produce polished writing or give engaging talks without explicit coaching on how to do this — particularly in a mathematical context.

In this post I want to focus on what I’ve done to improve students’ skills in presenting completed work. These ideas build on and adapt (in places, minimally) the great structures that my colleague A/Prof Lesley Ward has put in place in the Mathematical Communication course in our undergraduate mathematics degree.

**Brainstorming characteristics of good and bad talks**

Students have heard many talks, likely more bad than good! I start by asking them to brainstorm in their groups the features of talks they’ve heard. They write them on the whiteboard under the following headings:

**Delivery:**volume, rate, articulation — how the message is transmitted.**Language:**choice of words, grammar, using highly effective phrases — how the message is conveyed.**Organisation:**sequences and relationships amongst the ideas — how the content is structured.**Content:**what is said and how it is adapted to the listener and the situation.**Visual aids:**slides, physical props — how the aids reinforce the major points and stimulate the audience.**Other:**any ideas that don’t fit elsewhere.

There is now a collective understanding (or at least appreciation) of what to aim for and what to avoid. After this, I hand out the final presentation rubric which will be used to score the final project talks. This handout is minimally adapted from one used at Harvey Mudd College.

(Large whiteboard images here and here.)

#### Structured skill-building

Each week we start one class with a mini-talk (the other class starts with a Visual Pattern; thanks Fawn!). The aim is to slowly build skills, and to gradually improve. Talks increase in duration, and in other requirements.

**Mini-talk 1: Getting started!**A week in advance of the first talk, students choose their talk partners (usually a friend). I also hand out a mathematical fact (Collatz Conjecture or Koch Snowflake) with a few ideas for what to talk about, but they need to research some more themselves. They deliver their three-minute talks at the desks. After the talk, the listener thinks of one thing the other person did well and tells them. All elements are designed to be the least threatening option that I can think of.**Mini-talk 2: Build confidence; make improvements.**Students choose different talk partners a week in advance. They re-present their fact; the aim is to revisit what went well last time and what could be improved.**Mini-talk 3: Standing up; using whiteboards.**I select student pairs from each student’s usual work group. Students give three-minute talks about their project topics at the whiteboard.**Mini-talk 4: Present a new topic; listener steps up.**I select student pairs from among the class.**Mini-talk 5: Preparing visual aids.**I select student pairs from among the class. Students need to prepare something beforehand to write, draw or show. They present the ‘other fact’ from Mini-talk 4.**Mini-talk 6: Putting it all together.**Self-selected groups of three. Students give four-minute talks about any mathematical topic of their choosing (not their project!). Must use visual aids to engage their audience. Students self-evaluate afterwards according to the presentation rubric. Students also practice time-keeping and ‘not panicking’ when shown the ’30 seconds to go’ yellow card.

I haven’t talked here about guiding students in selecting which parts of the mathematics to draw out and which details don’t need to be told; how to tell the mathematical *story *is an important skill to develop that I’m not going to cover in this post.

You can get a PDF of my guidelines here. What I’ve set out can of course be adapted and expanded; the key is continual skill development and exposure to giving talks. By the time students get to the final 10-minute presentation they are fairly comfortable with giving talks about mathematical ideas to their peers. The final project talks are also far more enjoyable to listen to because of it!

What tips do you have for developing presentation skills in students? I’d love to hear about your strategies.