I am a Phd student at UT Austin, currently teaching/TA-ing various math courses here. I have taught math for 6 years now, both individually as well as in small groups and in a classroom. My office is always packed with students coming in for help because they say I explain it better than the professor. I even have other TAs' students asking for my help.
I usually figure out how a student thinks fairly quickly, and after that, it is easy to adapt to their mode so that it's almost as if they are figuring it out on their own. I can usually foresee the mistakes they are about to make, and by the time we are finished they can foresee them too and hence are able to avoid them.
In my experience one of the main reasons why people can't succeed in math is that they are just too afraid of it, and this compromises their ability to learn it. My goal is to get rid of this fear by making the material as simple as possible. Once this is established, students are able to understand and reason through problems much better. In some instances they even begin enjoying it; in fact, a few of my students who used to hate mathematics ended up majoring in it in college!
I believe that the right teacher can get any student to succeed in mathematics; it is simply a matter of acclimating to the student's needs. This is exactly what I am good at and why people keep coming to me for help. If you need help with math, I can get you back on your feet!
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