This is an interesting puzzle, for sure! A few thoughts come to my mind.
The Joy of Peer Review
I'm a huge fan of having students review each other's work. All the research shows that if done properly, it is one of the most effective ways to reinforce existing knowledge and deepen learning.
Since you aren't working with these students in a group setting, this becomes more challenging. However, I still think it is possible. Make copies of their work as they bring it to you or take pictures of what they write on a whiteboard. Then present one student's work to another and say "so and so was working on this problem and came up with this solution. Do you agree with it? Why?"
Since our educational system places a high emphasis on original, independent work, I would recommend telling your students why you are doing this beforehand. My favorite talking points include:
- Most "real" mathematics is done in groups. Adult mathematicians who discover interesting, cool things don't work by themselves.
- If you are researching a new topic and find a solution online, you need to be able to understand if it works and point out any flaws that it has. This helps everyone do better math.
- Thinking about work from this angle provides different exercise for their brains. Athletes don't only do one kind of work out and expect to get stronger. Flexible thinking will make them see different things.
Present them with a wide variety of other work. Make sure to give them correct solutions as well as incomplete or incorrect ones. You can also give them a correct solution but where work was not clearly shown. They can learn from all of these examples.
When they spend 30 minutes trying to understand why someone else's function doesn't graph the way that it should only to find that the sign was flipped, they will be more diligent with their own work.
Let Them Find Their Own Mistake
In addition to letting them find each other's mistakes, you can let them find their own just like they would if they were editing a paper for an English class.
When they come to show you work, don't instantly tell them whether their answer is right or wrong. Instead, ask them to prove its correctness. Again, you can cite the idea that mathematicians who do interesting work aren't working on problems where the answers are known already.
Ask them how they can prove it to you/themselves. If they aren't sure, you can offer suggestions:
- Graph it!
- Explain how they reached the answer
- Solve it "backwards"
If you have a chance to gather several students together, you can have them "present" on problems and encourage them to ask hard questions, point out problems in each other's solutions, or express confusion with the way something was explained or proven.
They will get into a habit of checking their own work, which will make them even more independent as learners and will eliminate this sort of problem.
Good luck! It sounds like you have an awesome group of students!