The problem with most graduate assistants is they're great at math, but have little or no idea on how to actually teach. There's this huge misconception that being good at a subject automatically makes someone qualified to teach said subject. Nothing could be further from the truth. Doing math and teaching math are two very distinct skill sets (not mutually exclusive, obviously).
The biggest problem graduate students have with teaching undergraduate math is not being able to comprehend "not understanding math." What's obvious to you will not be obvious to them. Not all students learn the same way. The way you were taught may not be the best way to teach.
The lower level classes most likely will be populated by students who are not mathematically inclined. Teaching under the assumption that they all are will just lead to disaster for the majority of them. But don't misunderstand me: know your learning objectives, and have your standards. The trick to teaching these classes is to have a certain level of expectations, but be prepared to reach down to students below that, and work to elevate them. If all you do is fly over their heads, and never swoop down to help them fly higher, then the students in your class are wasting their time and money.
Frankly, I think it's unfair to relegate the teaching of lower level courses to graduate students as a necessary part of their scholarship/fellowship/etc.. It's asking many graduate students to do a job they haven't been trained to do, and for many undergraduate students to spend their time and/or money for a service being offered by an untrained professional.
I'm not saying graduate students can't be good teachers. I've seen some great teaching done by them. But I've seen far more try to teach it as if their students were math majors when they majority of them were far from it.
You want some advice for teaching this level of math? Instigate dialogue in the class. Monologue is very boring, and turns students off, which is counterproductive to your goal of educating them. If students wanted to just listen to someone talking for 45 minutes, 50 minutes, 75 minutes, whatever, they can just search YouTube for math help. If all you do is talk for the entire time, students are using their ears (maybe) and eyes (maybe) as sensory receptors. But if there is dialogue, then students must produce to participate, using much more of their brain. This increases the chance of retention.
This leads to the question of how to instigate dialogue. The answer is the question, literally. Ask questions. And I don't just mean "Does anyone have any questions?" or "Do you agree?" or "Is this right?" or "What is 2+2?" (or any other math problem with a short answer). Short answer questions get minimal involvement on behalf of the student. The types of questions I'm talking about are more open ended and require a more in-depth reply. Any question starting with "Why" or "How" usually is good.
Another thing you should establish early is for students to not be afraid to give wrong answers. There is always something to be gained from a wrong answer. If you ask the student to explain an answer, it let's you get into their heads to see how they think, and as importantly, where they are misunderstanding something. Identify that gap, address it, and it not only benefits that student, but the whole class (because who knows how many other students were making the same mistake).
The crazy thing is, at the same time, you also have to be aware of how much you need to cover in a given day, and keeping the class on pace to get there. It's a tricky balancing game, but one that becomes more intuitive as you teach more and more.
Another thing to consider is this: do you want to tell them in advance what they will be learning that day? Many, MANY teachers start the class by writing a list of learning outcomes on the board, a sort of laundry list of things they will earn today. I always have such a list, but I never start the class that way. Why? Because nothing motivates it. I always start by posing some problem we want to try to solve, whether it's an applied problem or a theoretical problem. That problem motivates the math. Of course, you should know the path towards the solution (or rather, a path, as there is often more than one). Guide your students from beginning to end, allowing THEM to develop the math as much as possible. Well-worded questions can guide them down the path to discovery.
Not every student will like this. Many want you to just tell them what to do and how to do it. But guess who takes ownership of that knowledge when it's forced down their throats? No one.
But, if they build it, if they discover it, if they summarize it, then guess who owns it now? Every student involved. I even name rules, formulas, theorems after students who verbalize them (even if it seems trivial to me). Because that brings them in, and they want more.
I tell my students on day one that some of them are not going to like how I run my classes, because I don't just tell them what they are expected to learn at the start of each class. But I guarantee them that by the end of class, they'll know exactly what they are expected to know. Some students don't like the fact that I ask more questions than make statements. But I always make sure at the end, things are tied up in a neat little package: what you learned, rules, formulas, whatever.
It takes a while to internalize the art of teaching by asking questions. I was lucky. Between undergraduate school and graduate school, I worked for three years with a company that taught math to public school kids by asking questions. For three years, I wrote daily lesson plans that were nothing but questions. I never used it as a script during class, but it helped me prepare mentally for how to teach by questioning. And 25 years later, it's still my go-to approach in class (although I don't need to plan nearly as much in advance; most of it just comes naturally). Want to practice? Pick a topic you'll cover in the first week of whatever class you are teaching. Figure out what you want them to learn by the end. Start with a sample problem. Then try to come up with a question sequence that will lead them down the right path. Anticipate wrong answers, and how to get them to see their errors. Ask, ask, ask. (I should admit that yes, sometimes you have to step in and tell them something. After all, you are on a time constraint.) Make your classroom alive with dialogue as much as possible.
I've had many colleagues tell me this is a waste of my time in the classroom, that it's quicker and easier to just tell them what to do. And they are right about that: its quicker, and it's easier. But that places emphasis on quantity over quality. You need to ask yourself what is more important: rushing to cover topics A, B and C in a fixed time interval, or investing more time to have a higher quality class experience at the risk of falling behind? I always choose quality over quantity. If towards the end of the term, I see time is running out, then I will default to standard monologue for the sake of time. But I will always choose quality first.
Ok, this was much longer than I initially planned. But I'm just a tad passionate about teaching college and university students. To take students who are convinced they can't do math well (and trust me, in your lower level classes, that will be the majority of them), and have them confidently do that math at the end of the semester, makes it all worthwhile.
Unless you are my graduate school advisor, who told me I was an idiot for focusing on teaching so much. Great mathematician, sucked at teaching. To each their own.
