Learning involves a process of acquiring knowledge by focused attention, observation, and practice, and then eventually forming, integrating, and using concepts as part of the student's full body of knowledge. Early in a child's life, a rudimentary form of learning may be accomplished through rote memorization and repetition of facts. Truly understanding a subject, however, requires a student to focus on the content and the meaning of a given subject, to isolate its essentials, to establish its relationship to what the student already knows, and to integrate it with the appropriate categories of other subjects. An education that trains a student's mind to think, therefore, would be one that teaches him to make connections, to generalize, and to understand the wider issues and principles involved in a given topic. The ideal teacher would achieve this feat by presenting the material to him in a calculated, conceptually proper order, accompanied by the necessary context, as well as the evidence that validates each new piece of information.

Math is uniquely suited for demonstrating the consequences of such a theoretical approach to learning. It is my responsibility as a mathematician and math teacher not only to teach my students the concrete, factual knowledge of a geometry or calculus textbook, but also to examine with my students what this information communicates about ourselves, our world, and our culture, carefully balancing concretes and abstractions, identifying and explaining the reasoning at each point in the lesson, and preparing the students' minds for drawing connections between mathematical concepts and the world outside the classroom. Indeed, these are the processes that must first be absorbed each academic year by the student in relation to many different subjects. Upon graduation, such training will encourage an attitude of self-confidence and self-efficacy in regards to learning.

My primary function and goal as an educator, therefore, is to help to instill in my students the skills and mental habits necessary for them to learn anything they want by themselves, which means, to be interested and excited about their own intellectual development.

My overall strategy for ensuring that a student understands the material may be summarized as follows: for a student to learn, he must "hear it," "see it," "feel it," and "repeat it".

(1) "Hear it." During my years as a college freshman, I remember feeling surprised when professors occasionally assessed students on material that was never covered in class. No one can expect students to learn material that their teacher doesn't bring to their attention by informing them about it in the first place. The first step in teaching any material is that students must hear the material presented stated to them aloud. Since no two minds are exactly alike, I anticipate encountering a variety of learning styles among my students. Thus, I will need to utilize a variety of ways of communicating the same material to the class-that is, of relating to a wide variety of individual students with a wide variety of backgrounds--so that no one is confused and nothing gets lost in translation.

(2) "See it." A math teacher worthy of the name must continually show to students how and why math works. He must give them evidence; he must help them to see real-world applications. He must always be ready to answer the perpetual question that students rise: Why are we learning this? What is this good for? Any way of presenting math in the real world other than, and in addition to, markers and a whiteboard will be immensely helpful.

(3) "Feel it." This dictum has two separate meanings. (I) Anything that a teacher can do to provide students with a tactile experience involving the material is immensely valuable. Hands-on activities and lessons that involve interaction between the student and the material are key, especially at the high-school or university levels of education. Students sit at their desks for significant portions of the school day; thus, anything I can do as their teacher to get them out of their seats and moving will improve the learning environment, especially when this allows the student to solve a problem or explore a concept for themselves. (ii) "Feel it" also speaks to a successful teacher's passion for the subject. In order for my students to get excited in the classroom and become proficient at math, they must first sense my genuine passion for teaching and for math. Carl Sagan said it best when he was speaking about science: "When you're in love, you want to tell the world." In the same way, I see nothing wrong from letting that passion show in the classroom. The delight I get from doing math and talking about math should be contagious. Such a disposition will encourage wonder and curiosity in the students, for if they do not see why their teacher is excited about math, they will at least be more likely to pay attention in order to discover what he sees in the equations and diagrams that fascinate him.

(4) "Repeat it." It is not enough know the theoretical basis of a subject; a student must be able to put theory into practice and demonstrate that they understand how to apply theory to the real world. In math class, it is not enough to read through a textbook or a proof in order to learn the material; students must repeat the action they have seen performed in class to learn it, through homework, problem sets, and formative assessments. Deliberate practice is vital for a student to internalize the material I try to share with them.

For a typical tutoring session, I always bring with me pens, pencils, notebook paper, spiral-bound notebooks, and blank note cards as raw materials for working out problem sets and recording material for further study. I always make sure to devote one section of a blank spiral-bound notebook to scheduling and planning ahead for the student's upcoming tests and assignments, as well as to coordinate further tutoring sessions. Whenever possible, I always require the student to bring their recent homework assignments, tests, the class syllabus, and books used in the class so that I can understand what the student's teacher has already taught the student and glean some clues as to where that student is struggling.

During the first day of tutoring with a new student, my first priority is to build a rapport with the student, as I want to be someone he or she feels comfortable talking to, and first impressions are key. For that reason, I will not ask about math or other academic subjects initially. I will ask about life at school, interests, and other colloquial subjects for the first five minutes or so. I want to let the student know that I am listening to and learning about him or her as an individual. I can also discover what motivates the student during this conversation and plan for how to frame future tutoring sessions in terms of what the student already knows and enjoys.

When we talk about math, I want the conversation to be very general. I want the student to tell me more about their schoolwork, but on the first day I am primarily listening to how they speak about their schoolwork so that I can judge where they're already comfortable, where they're struggling, and what kind of learning style they intuitively use (i.e., are they visual learners, auditory learners, or kinesthetic? Do they learn best by observing, listening, or doing?). In this way, I can also infer how the student views him- or herself as a student; that is, his or her self-concept in relation to schoolwork. We may work on one or two specific homework problems that the student does not understand, but I will continue to redirect the conversation to general principles of the underlying content material, so that the student has a better grasp on the mathematical context and methodology behind those specific problems.

I don't necessarily want to help finish the student's assignment on the first day, although it's likely that we may do that anyway; my priority for the first session is to set the proper context for future learning and for underlying concepts, both about the content areas in which they are struggling and themselves. The most important thing the student needs to know at the end of the first day is that we can work together, and that they are capable of far more than they think they are.

Because I truly care about the intellectual development of my students, I encourage and inspire as far as possible, habits of auto didacticism in the subjects my students want to explore, as well as within the classes they have to attend. I'm not satisfied in most cases when my students only want to achieve a high grade in their courses; I'm saddened whenever I see a student's curiosity permanently dulled by their school's emphasis on grades, results, and regurgitation of memorized material over gleaning personal satisfaction and demonstrable skills in their efforts to learn. Therefore, during my sessions I emphasize whenever possible the immense value of learning how to learn rather than what to learn, so that my students may be motivated to retain a love for learning after school.