Internal testing first
Make sure you have run the game yourself several times with the materials you intend to bring. This will help you ensure that you have all of the physical pieces you need.
Check your vocabulary
It can be challenging as an adult to remember what you didn't know when you were ten years old. Don't trust your memory: go carefully over all of your key terms, perhaps with kids, teachers, or online aids, to ensure the vocabulary is age-appropriate. As a counterexample, in our Morgan's Raid game, we used the words "chaos" and "reputation," but it wasn't until playtesting that we found that kids' understanding of these terms was much different from our own—so much so that it affected the learning outcomes!
Dry run rules explanations
It's one thing for you to know the systems of the game; it's yet another one for you to be able to explain it to someone else. Dry run complete rules explanations, possibly with a friend, possibly with a rubber duck—surprisingly, it may not matter which you choose. Don't just practice what words you will say, but also consider how you will use the materials to demonstrate gameplay. Will you show a sample turn? Will everyone participate in a sample turn? Is the game itself easy enough to get into that you can explain as you go and maintain a sense of fairness?
If you don't have a helper or producer over your shoulder to collect playtesting data, you may be stuck both explaining the game and evaluating the session. Make sure to take notes as you go, and write yourself an analytic memo immediately after the session. Don't trust memory: write it down. If you do have someone else who can observe and take notes too, all the better for triangulation!
Watch body language
If the kids are leaning in, they are engaged. If they back off, they are losing interest. This makes sense, right? What's fascinating about it is that the players themselves probably don't know they are doing it, and if you ask them to explain it, the rationalization part of the brain will kick in and make something up. Keep an eye open for body language: it tells you what's really going on in the players' minds.
Plan post-play questions
Open-ended questions like, "What did you think?" are often unproductive. Think about the design goals of your game and the design questions you still have in mind, and plan questions that will help you get at them. If you're making an educational game, use this opportunity to practice a debriefing, seeing if the players can connect their gameplay experience to the learning outcomes. Drawing again from Morgan's Raid, we used the question, "Do you think General Morgan was a good guy or a bad guy?" We had some amazing responses as different players voiced their opinions, and they then realized that there wasn't one objective answer, and they had to start thinking about perspective and context.
That being said, there are two great general-purpose questions that I often use. The first is, "What would you change about this game?" The trick here, as is often the case with playtesting, is that you are not really looking for design advice: you are looking for the areas the players are point to as weaknesses. That is, their responses may be signals to areas that need revision. The second is, "Do you have any questions for me?" This is a great closer that I use in playtesting and in interview protocols, as it empowers the players to voice their own thoughts and concerns.
Thank the players
Don't forget to thank the players. Let them know that by playtesting the game, they have contributed significantly to your work and your studies.
Anything I missed? Please feel free to add more tips in the comments!