Probability

Probability is a topic which lends itself well to interactive demonstrations, and is popular with many audiences because of the excitement associated with gambling, and the personal interest in health-related risks. The problem with this is that probability goes wrong a fairly predictable amount of the time, so you *need* fallbacks in case your demos don't work.

You might be able to redeem things by explaining *why* things have failed - but you need to frame it so you don't lose all credibility!

A particular issue with this topic is many famous interesting probability 'problems' are based on common misconceptions, which might be deep-seated and hard to challenge. The misconceptions can come from quite subtle points which need careful unpacking. A good example is the Monty Hall Problem, in which - for example - people often fail to point out that when Monty picks a door to show you behind, he's doing so deliberately so he will always show you a goat, and not just choosing one at random. Be careful you don't end up reinforcing an unhelpful stereotype of "probability is just confusing".

You're often limited in how much time you can spend setting something up, but think carefully about your explanations, as stating the setup wrongly can change the entire problem (and the answer!) Rob Eastaway has written a blog post about the 'Boy born on a Tuesday' problem which outlines some of the issues.

Proof

Of all the abstract topics in maths, most teachers will tell you that trying to communicate the nature of 'mathematical proof' is one of the hardest topics to get across. It's worth remembering that, at its heart, proof is among the most 'human' aspects of mathematics - it's about communication rather than correctness. It's about *convincing* rather than *calculating*.

Most human beings want to be convinced if they see something suspicious or dubious. This means that a demonstration of something surprising can motivate a proof very nicely:

Use a picture

Generally, more abstract mathematics is harder to communicate than something visual like geometry; talking about higher-dimensional spaces is a great way to stretch peoples' imaginations but you need to make sure the audience can follow what you're talking about. Introducing notation for abstract ideas is what mathematicians do, but it takes time to get used to this. As a maths communicator you might need to revert to a picture of an analogy, rather than trying to drag an audience through the painful process of getting used to new vocab and notation.

Applications

Applied maths often gives you nice real-world examples to use, but it's important to make sure you're still communicating the underlying maths when you do, and not just ignoring it in favour of the surrounding context.

Research level maths is very difficult to communicate compared to research-level science; there are some branches of maths which are very pure and abstract, and unless your maths has obvious real-world applications you’ll need to work to draw analogies and find concrete examples.

Many abstract maths ideas rely on pages of crucial definitions and scaffolding, much of which you won’t have time to build in a talk, so you’ll need to think carefully about how you can simplify without what you say being false. ‘Subject to some other fine details’ and ‘for well-behaved examples’ cover a multitude of sins.

Presenting Data

Communicating about research data and statistics comes with its own pitfalls - the uncertainty inherent in this type of information means it's possible for people to get the impression you aren't sure, or that what you're saying is unreliable. This blog post by Rachel Thomas and Marianne Freiberger outlines some ways to make sure you're clear and trustworthy.

Journalists often need to communicate data in visual ways, and the Financial Times has developed a 'Visual Vocabulary' (PDF) comparing different types of visualisation, and their relative strengths for presenting different types of data.