I am writing to share some of what I have learned over the past few years. As I’m sure you will soon learn, this job is much more challenging than assisting with lower-divisions courses. You will, I assure you, not know everything, sometimes nowhere near it, yet the material is interested and you are a much more valuable resource for the students than a tutor often is for an introductory level course. Also, the students are more committed and tend to be more prepared and engaged with the material. Your job is different as well. You are no longer really helping students learn how to solve an individual problem anymore or helping them memorize a single technique or argument.
In my own academic career as a math major, I realized that the most effective tool in upper-division courses is a solid understanding of logical arguments, a willingness to play with approaching problems in many different ways, and the perseverance to attempt solving a problem for hours, sometimes without success. These are the skills that I hope to pass on to my students. Certainly, if there is a specific topic that they struggle with, we will go over the topic, but more important than that, almost no two problems will be exactly alike so they need to be able to think creatively and be able to apply the underlying concepts. This is where it gets tricky. How do you teach someone to be able to play around with ideas until it works, yet selectively choose the ideas that you would likely want to try first? I find that this is often what I spend my time doing.
Instead of emphasizing the solution itself to the problem, I focus more attention on why I choose to approach a given part in a certain way. While the students often just want to get their homework done and might hope that you finish your long talk on why you chose to do the first step first and the second step after, having their homework done will in no way ensure that they understand any material in the course, and likely more importantly for them, that they pass the course. So I have found really speaking through the logic behind moving from step 1 to step 2 is both necessary and challenging.
Another challenge faced is the massive amount of material you will likely be forced to recall. You won’t know how to approach every problem or recall every topic, so I make sure always to bring my own notes (unless I know the material really, really well) to tutoring so that I can reference the hypothesis to a particular theorem, or check my own homework solutions if I used their same textbook. You won’t know everything and they will understand, so don’t worry if approaching a problem in a certain way fails even if you spent twenty minutes trying to make it work out. The students understand where you’re coming from, but make sure to point out that it’s not a bad thing to approach a problem one way and have it not be successful. Use it as a teaching point. You did well in the course and you can’t answer every question the first time you try it, so they don’t need to give up on themselves if the problems don’t work out the first time.
The students you are working with will be different as well. There will be a lot of variety in academic preparedness. Maybe this is a students’ first upper division course and they won’t be able to finish their homework assignment in an hour anymore. Make sure to encourage them to start early and use all of their resources as there will probably be a lot less of them. With this decrease in resources (fewer sections, only a single TA, etc.), the students tend to value the sessions more and seem to be more respectful of the tutoring schedule. I have found that there are many less no-shows or students who continually say they have no questions than when I tutored for introductory courses. The consistency in attendance is good, but I found a substantial increase in my large group sessions, such as having primarily 4-5 person sessions and almost never a session with only one or two students. Tutoring this large number of students (I worked with over 20 every quarter this year) makes it challenging to know their individual abilities. So don’t let any students get away with sitting quietly in the corner every session. Make sure to engage all of them and check in to see who gets stuck where. You will definitely be more effective with shy or completely lost students if you force them to show you where they are having difficulties. They will hopefully be thankful for the clarifications.
Overall, I would say you will learn by doing. As long as you have good knowledge of the material and a desire to work with the students, tutoring will overall go really well. The students will understand when an idea was the incorrect one and will still find their time with you valuable. Good luck with everything and I’m sure the students will be glad for all the extra assistance you will provide.