My philosophy is that anybody is capable of learning anything; they need only have the will to learn it and the proper teacher to guide them.

I would introduce myself, go over the basic components of the subject, ask what they know on a more advanced level if anything, walk them through the techniques they need to succeed on their own and see how they handle themselves. I would adjust myself and my teaching in future sessions to best accommodate the student based on this.

When I have finished teaching the student what he/she needs to know on the basic level, I task the student with solving problems on his own, only stepping and helping (if needed) at the very end. Further, I may even ask the student to teach me what I just finished teaching them, which would truly demonstrate a real understanding of the material.

I typically try to get on the same level as my students by relating to them as best as I can. I am still fairly young and "with it" myself, so I think that I should be able to understand what may be demotivating them. Regardless, a good sense of humor and positive attitude can go a long way to getting students motivated in lieu of that, both of which I like to think that I have in spades.

The key to teaching a difficult problem always lies in how much that you present at one time. Even the most advanced Calculus can be broken down into small enough pieces that can be taught to anyone. So essentially the idea would be to look at the problem, break it down into smaller and smaller pieces until the student can understand all of them, and then essentially putting them back together to form the core understanding of the concept.

I try to introduce myself, get to know the student, make a humorous and approachable environment in which they feel comfortable to work in, and generally be equitable and understanding of the difficulties that they may be having with the subject matter.

To be perfectly honest, sometimes there really is no way to get excited for certain subjects that you may not like or have a lot of trouble in. And that's totally okay. We all, myself included, have those subjects that we really will never particularly like, but the important thing to recognize is that we have to do them anyway. That's life sometimes. I suppose the way to "get excited" in those scenarios is to figure ways to efficiently tackle the subjects in as painless a way as possible, realizing that it will be over soon enough, and doing well enough that you won't ever have to do them again.

As I have said previously, the ways I would make sure that a student understands material is by both having them do practice problems for me on their own, and even by having them attempt to teach me the subject material that I had previously just taught to them afterward.

I think that confidence will be built naturally through accomplishment. The most important part is that the student never feel as though what they are facing is insurmountable. If a problem is too hard, simply by breaking it down into easy to digest pieces, the student may suddenly find it easier and have more confidence approaching problems like it in the future. Regardless, it should be emphasized that I feel that the student is capable of learning; it's as simple as that.

By asking them. That's probably the easiest way. There obviously is a more complicated layer to this aspect as well though. Typically, if a student is unsure, I will ask what they do/do not know about the subject, and if they are able to solve an example problem for me, I can further diagnose what I need to teach.

I typically only use what is necessary in order to not overwhelm a student, though that can vary based on the complexity of the Mathematics. Basic Math only requires some scratch paper and a pencil, Math in geometry and trigonometry typically benefit from graphing paper and a scientific calculator, Calculus requires a graphing Calculator, etc.

I will usually have a student read me a passage, tell me what they think are the most important takeaways from the text, and underline those sentences which support their claims to come back to later. Beyond that, reading comprehension can only be really improved through a lot of reading, and critical thinking based on said reading.

Again, I believe that the best way to adapt to what a student needs is simply to ask them what they need. It is not only helpful to me, but also helpful for the student to visualize what it is they need help with; i.e. the goals that they need to set for themselves. If there is a scope of needs that are unbeknownst to the student, it should reveal itself in the lesson/practice work. I make sure to draw special attention to these things to make sure that the student understands exactly where he/she is having trouble.