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What’s the best experience you’ve had with a math TA? What made them so helpful?

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This is probably on the low-tech end of the spectrum of answers you're going to get, but the best TA I ever had was in a prob/stochastic process course, and the best thing he did was (either subtly or explicitly) treat every question we brought to him as though it was an important question to ask. There's always this fear that you're going to expose gaps in your intelligence by asking questions, and this guy was able to completely remove that barrier to asking questions by the way he responded to them.

It made the people asking the questions feel more validated, and gave an additional confidence boost to the people who already knew the answers - because it meant they knew the answer to something important.
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I've had TAs who were better than the professors leading the class. Usually their best qualities were being able to see the subject from an undergrad level, rather than way up in the clouds in a way that was powerful and interesting but ultimately inaccessible for undergrads.

Its always great when anyone teaching maths, especially when going through work students have done, calls out the important concepts in a problem. Showing me how to do a proof I couldn't is great, but pointing at the crucial step that I'm bound to see again -- and not getting too caught up in specifics of the problem --is much better.
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The best math TA I ever had (in a graduate-level Algebra class) did two things perfectly:

(1) He created an environment where we all felt comfortable admitting that we didn't understand things (one memory that sticks out - at the start of the semester, he went around the room and asked us each to say, individually, "I don't understand" - he said then it would be less scary next time we had to say it)

(2) He pushed us to explain our thought processes as much as possible, so that we could see for ourselves whether or not we really understood something. And since the environment in the room was so comfortable, it wasn't stressful or scary to get halfway through the sketch of a proof before realizing that you didn't know what to do next. When that happened, he'd get excited, and it would start a discussion.
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I think one of the biggest things for any subject is being able and prepared to explain a concept in a different way than is "conventional". If you explain a concept, and the student doesn't get it, going through the same explanation slower won't necessarily help, but coming at the concept from a different direction can be huge. People just think differently!

Also, I would suggest not referring to anything as "easy" or basic. The first TA I ever had was doing a recetation and skipped some slides because it was "basic and we should already know it". I didn't know it, and it really discouraged me from engaging from the class and asking for help. Of course this is not necessarily great for classes that have prereqs, but if it's content you can't be 100% sure they covered in previous classes it's probably worth taking some time on.
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First day of class for DiffEq and our teacher asked everyone to spend the hour on two questions. The first was to write down the Fundamental Theorem of Calculus and explain it in our own words (noone could remember the exact wording), and the second was to write an essay, as long or short as you felt necessary, on your own perspective of the nature of mathematics. He gave as a basic example whether you thought mathematics was created or discovered. This not only gave him perspective of where each student stood academically, but fostered further interest in the philosophy of mathematics. Each day of class began with a group discussion regarding the prior lesson, and he would not move on until everyone understood. He was by far my favorite uni math prof.
For anyone interested I was (and still am) a Neo-Platonist but was experimenting with Intuitionism. This was before I read Russell, Godel, and Cantor though. It was his class that led me to do so.
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Probably the one that has the most impact on my teaching is that when a student asks me for help on a problem, I generally start by getting them to explain their thought process or show their working, or what they understand about a relevant definition/concept e.g. "Before we jump into this, what do you know about <concept that the problem uses>?".

It puts the agency onto the student and gets them to mentally load up the problem and relevant concepts in their head so that they're in the right headspace to address it, and thus makes their opinion and thinking feel valued because you're showing interest in it.

Note that 'asking for help on a problem' is different to when a student asks a question, as some questions tend to be pointed and specific in which can be answered directly. If the question is a bit more open ended or lacking context, then you'll want to ask about the context first.

In terms of feedback, avoid any kind of emotive language or negative adjectives. It's very easy to let those slip through when you're e.g. having a frustrating evening grading shitty assignments. Just point out the mistake as normally as having a conversation, and/or make pragmatic/realistic suggestions on how the student can do better (e.g. 'Study harder' is unhelpful!).
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I was late grading homework one day, so I told my students the following.

"""
Sorry I couldn't get your homework graded. Terry, the tarantula, got out of his cage last night, and he got into a fight with Larry the Lizard. I had to take Larry to the vet last minute and was unable to complete the grading.

I'm sorry.
"""

As a graduate student, the only pets I had were the cockroaches living in my apartment. I usually made dumb dad jokes while TAing and expected people to act in disbelief, but instead, I learned that I looked like an exotic pets person.
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Lots of big picture math stuff, so here are two smaller things:

- you can build trust by doing the administrative part of your job well. E.g. always bring that folder of homework (if applicable), know the dates of upcoming exams, etc.

- if you are a TA for a bad instructor then it's tempting to badmouth the instructor. Resist this and redirect student complaints to math where possible. E.g. a student says "yesterday's class was so bad, he didn't explain any of the examples" you can respond "ok let's look at one of those examples again." This also builds trust -- most students want to get a good grade more than want to talk shit.
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Darn I was hoping to see myself mentioned here.
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While both having TAs and being one for several classes, I had found that the two rules for effective TA-ing is to (1) Not give out the answers and prompt investigative thought, (2) Make them feel like every question they ask is interesting, and (3) Do your best not to be presumptuous about what students retain from prerequisites.   


First off, it will be tempting to fill a void of silence when prompting discussions or during one-on-ones. Every student will want to jump straight toward the solution and see if they can draw the connecting lines later. Prompt them to think. Ask tricky questions to tease their understanding and make them consider factors they might be missing. See if they can apply that logic to the main problem. (*For instance, I was lecturing about the isomorphisms that can be drawn between physical systems like a damped RLC circuit and a frictional mass on a spring. They were having a difficult time understanding the factors and concepts in the RLC circuitry equations, so I distilled it down to general behaviors like "what is losing the energy in the system", "what here behaves like inertia, resisting change", etc*.) If you equip them with the tools to solve the puzzle, all you have to do is reframe and tease until they understand.   


Second, asking questions is daunting. Be friendly, be respectful, and acknowledge how interesting a question can be. Students who feel comfortable asking questions inevitably ask amazing questions in class. You're in this field because it is a fascinating field. Act like a question asked to be re-explained for the hundredth time is just as interesting as the first time. Find relatable examples, invite them to ask more, and always follow-up. I often started these sessions out with, "Now, if you think you're missing something or are trying to remember something form a previous class, there are half a dozen others in the same boat. Do them a favor and ask."  


Lastly, and what I've found to be most important, is not to presume what students have retained from previous classes. I've had students who have forgotten basic trig identities, major assumptions behind statistical tests, etc. Check-in constantly, ask, "Does anyone need a quick recap on L'hopital's rule?" "Is this feeling a bit rusty to anyone right now?" Communication is important.

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