You may be used to giving talks as part of research seminars, or teaching degree-level maths in a university context. While these use many of the same skills, it’s important to remember that planning an outreach talk should be very different in some ways, and requires a different approach.

The main difference between research talks and outreach talks is the audience. An outreach talk might be to a group of school students, an audience of interested adults, or a mixed family group at a science festival. The main difference will be that you can’t expect the audience to have maths to degree level - school maths curricula are available online for you to check what school students of different ages are likely to know, and for general adult audiences you can reasonably assume knowledge is at GCSE-level or slightly lower (since it might be a while since they studied!)

There will be differences between e.g. a group in a school who are required to attend your talk and a group at a science festival who have chosen/paid to see you; there’ll be differences between e.g. students in September and the same students in June of the following year (a year 8 in September is effectively a year 7!) and you’ll also find within any given audience there are different ability and interest levels. You should try to get as much information as possible from whoever’s organised the talk to make sure you can prepare adequately - but be prepared for last-minute changes too.

It can be difficult to pitch a talk under these circumstances, but by asking questions of the audience, watching their reactions and giving them opportunities to contribute you can gauge this - especially in smaller group situations like table activities or busking - and tailor your explanations and even content accordingly. Generally though, err on the side of explaining more - people are surprisingly happy to hear something they already know, as they can feel smug about it, as long as you also include something new and more difficult.

One other thing that’s different about an outreach audience (compared to a seminar audience) is that they won’t have the same determination to learn, and you can’t afford to just talk for a long time as some will lose interest and become visibly bored, which will be apparent to others; even those who aren’t visibly bored might be hiding it well. You can keep people engaged by including chances for them to do things and interact, using engaging visuals/animations/demonstrations and asking them to think to themselves or discuss something for 30 seconds (even if you don’t then ask them to share answers!)

Engagement talks can benefit greatly from trying-out and practice, to hone the level of explanation and content. Imagine you’re explaining it to a member of the public with a non-maths background (it’s useful to imagine someone you know that has a general amount of maths knowledge and picture explaining it to them - or actually do, if they can spare time to watch your talk!). Watch out for jargon and terminology - you’ll need to define a lot of terms before using them, and this takes time.

One of the classic mistakes in planning public outreach when you’re used to seminars is trying to fit too much in the time. You’ll know from the organisers how long your talk needs to be (and this may differ a lot from a standard seminar!) so you can work out how much you can fit in, but be conservative. Anything might go wrong (especially if you have tech, or demonstrations in your plan), there might be interruptions, or you may need to ad-lib extra explanation if people aren’t following. Aim to leave slack in your timings and room for the content to breathe. It’s tempting to try and include everything, but you sometimes need to be brutal - nobody will know what you’ve left out, and running over time is miles worse than under-running by a little (and you can always leave more time for Q&A, which in public talks can be much more varied and interesting!).