I believe math is important for students to learn because it teaches students problem-solving skills, collaboration with peers, patience and perseverance, and how to learn from mistakes -- all of which are life skills I want my students to have when they graduate. In order to graduate with these skills, I challenge my students to think about and understand how the math actually works.
Many of my teaching methods are taken from Liljedahl's Building Thinking Classrooms in Mathematics, whose main idea is that if "students are not thinking, they are not learning." The graphic below gives a brief overview of each chapter of this book.
The main methods I focus on in my classroom are 1) Begin with a Problem, 2) Visibly Random Groups, 3) Vertical Non-Permanent Surfaces (as my current materials allow), 6) Answering Questions, and 10) Student Notes. I explain all of these methods to my students at the beginning of the school year and as we do relevant activities.
I realize this will be a very new and different teaching environment for many students. If you have any questions about my teaching methods and/or philosophy, please feel free to contact me via email or phone call and I would be happy to discuss!