Most of my experience as a tutor has been in topics related to those covered in this test. While I find the concepts are quite similar to those covered when I have provided tutoring for more advanced testing (like the Quantitative Reasoning portion of the GRE), I am very familiar with a softer approach necessary for students whom are new to the topics, for whom a greater supply of context will facilitate their comprehension and ensure their success.
Majoring in the Engineering Sciences in college has reinforced my command of Algebra. After an extensive study of the methods of Calculus, Algebra becomes sort of like your ABCs of higher mathematics. The most elegant aspect of mathematics for me is its empirical structure: the more you continue using every piece of it, the deeper you direct your study of it.
I believe most students will find the second part of Algebra to be a welcome return to ideas introduced to them before Geometry. Most of my experience as a tutor has been centered on the topics encountered in this subject. From my education in Physics, I have come to believe that essentially all mathematical analysis will reduce to the algebraic expressions studied in this subject, and I believe a sturdy base of comprehension of it will be fundamental to all students who seek a university education in Engineering of any kind.
Calculus is the natural language of the sciences--it's very foundations run parallel to most every significant discovery in Classical Physics. From my instruction in the construction of Hamiltonians to my studies in the methods of Tensor Analysis, I am well versed in the advanced techniques of analysis that students may encounter. I have also studied extensively in Real Analysis (or so-called Standard Analysis), affording me great familiarity with the theoretical rationale behind such topics as the Limiting Process, the Fundamental Theorem of Calculus, or the Chain Rule. I believe, while quite rigorous and initially intimidating, an informed approach to the techniques of Calculus will, during instruction, reveal to students the amazing potency of this rich mathematical legacy.
Geometry instruction for most students begins with strange objects like lines and triangles. Proofs often seem especially confusing. My expertise derives from my studies in the physical sciences, which require a frequent application of geometric analysis (Einstein famously needed only the Pythagorean Theorem to illustrate Special Relativity). I like to give students concrete examples of the utility of the techniques, show them why proving certain ideas is so important.
Most of the tutoring I have provided for this subject has been focused on the quantitative reasoning portion of the test. I find most students require instruction for the verbal section like a an athlete needs conditioning exercises--mnemonics to aid their vocabulary skills, practice analogies, a refreshment of primer concepts in proper grammar. To that end, my role is usually that of a strict coach, monitoring them doing their mental sit-ups and push-ups. I find most students are weakest in their comprehension of those topics covered in the quantitative reasoning section. Their majors of university study usually have not included much mathematics instruction. To this end, my background in the Physical Sciences excels as a preparatory credential. I can offer students numerous examples to illustrate the test topics, enriching their confidence by solidifying their comprehension through context.
My university education included the Hamiltonian formulation of Classical Mechanics, the Maxwell Field Equations description of Electromagnetism, and non-Relativistic Quantum Mechanics. However, most of the tutoring sought after by students in this subject will not require such involved mathematics. My approach is normally to engage students in more accessible examples of simpler models in Introductory Physics. The more popular AP subject examination attempts a broad rather than deep assessment of a student's command of the techniques of investigation of physical phenomena, and I prefer to interpret those techniques into something more intelligible to students than what they may receive during their lectures. The most common mistake we make upon our first encounter with the subject is to mistake it as a set of formulas that can be blindly applied to a situation. Physics seems to me more of a sturdy, strange, sometimes stunningly elegant set of new insights into the machinery of reality, an introduction to the intuition and motives of a wonderful creation. I like to think that this approach makes the subject more vital, more appealing, more useful than most of the topics students normally encounter in their education.
Despite my education in Physics, my thorough command of Calculus, my studies in Elementary Number Theory, my studies in Linear Analysis, my studies in Applied Partial Differential Equations, I have never forgotten the inspiring suggestion of renowned physicist Richard Feynman, who stated that one who truly understands even something so arcane as Quantum Field Theory can explain it to a 4th grader. I have never lost touch with the opening concepts of mathematical analysis. I think the beauty of the Real Number Line is the starting point for conversations with students that can easily relieve the germ of "math anxiety" that so frequently distracts them from future success in their studies.
Precalculus mathematics tend to confuse most folks because of the eclectic array of topics. I think the key to focusing a student's comprehension is to inspire them with examples, illustrations of the utility of the analytic tools and techniques that come from a mastery of the subject. My background in the Physical Sciences has equipped me with an effective and inexhaustible supply of these.
Trigonometry is, to me, the introduction of the potency of geometric analysis. At last, students will find those concepts from algebra and geometry unified into some type of analytic tool. From the mystifying exercises of Proving Trigonometric Identities to the seemingly odd habits of Modifying Periodic Functions, I expect students will sense an intimidating breed of techniques compose this subject. Here, my familiarity with methods like Fourier Analysis and its underlying theory (like Parseval's theorem and L2 convergence) excels at providing me the necessary insight to assist any student with any question that may occur to them.