Socratic Tutor Guiding Students Through the Math Maze

David J.

Algebra 1

I have a B.S. in Theoretical Mathematics which included the study of linear and abstract algebra. I have more than five years of experience teaching students algebra. Furthermore, I have deepened my understanding of algebra through studying the history of the subject. In my experience teaching students in Algebra 1 classes, I have come to the opinion that the title is misleading. For the curricula of such courses, which are now nearly indistinguishable in their standardization across the nation, consist of topics from multiple mathematical disciplines, not just what mathematicians consider to be algebra, which is quite unknown to the lay person. Rather, it consists of introductions to concepts from analysis, geometry, and probability, as well as algebra. The most important goal of any student in an Algebra 1 course is to gain a thorough understanding of linear functions. In order to be well prepared for future studies, it is necessary to know how to navigate between their graphical, functional, tabular, and verbal representations. Furthermore, one must know what motivates the importance of linear functions beyond the need to pass a course.

Algebra 2

When I took Algebra 2 in high school, I was intrigued by the sudden profusion of topics not even hinted at by my experience in Algebra 1 and Geometry. I found the chapter on trigonometric functions particularly eye-opening, for it was the first time I began to see beauty in equations. It was during the study of trigonometric identities that I began to witness the axiomatic structure of mathematics. I learned that the myriad of identities and formulas need not be memorized (a daunting task), for they could all be quickly derived from only a handful if one has confidence in manipulating equations. In my ten years as a credentialed, California high school math teacher, I taught several sections of Algebra 2. In that course, and all others, I strove to help students acquire an axiomatic perspective that would allow them to avoid mere memorization as the only means of being successful in math. I did manage to help some make strides toward that end, though the educational trends of the day did not afford me the time to enable them to develop a proper appreciation and understanding of mathematical structure. While studying to earn my Bachelor of Science in Theoretical Mathematics, I took such courses as Probability Theory, Advanced Euclidean Geometry, Linear Algebra, Abstract Algebra, and Real Analysis. all of which provided analyses of all the topics of Algebra 2 quite rigorously. It was during a course in the History of Mathematics, however, that I was treated to an analysis of The Elements of Euclid. Witnessing the earliest recorded, successful attempt at organizing all known mathematical knowledge was revelatory for me. It was then I realized my high school epiphany was really my having tapped into an ancient and highly celebrated thread of thought. I haven't been able to turn away from mathematics since!

Calculus

I found myself in AP Calculus on the second day of my senior year. On the first day, I had to prove that I successfully taught myself Pre-Calculus over the previous summer. While my classmates were initially surprised, because I had not previously inhabited their academic world, I went on to impress them with my problem-solving prowess and not least of all by scoring a 5 on the AP Calculus Exam! The credit from the exam score enabled me to skip Calculus I as the first course in my studies as a theoretical mathematics major. Since then I have studied many courses, such as Multivariable Calculus, Differential Equations, Probability Theory, Real Analysis, and the History of Mathematics. Those courses served to deepen my understanding of not only the logic behind Calculus, but also of its beautiful applications to celestial mechanics. Currently, I am studying Widder's Advanced Calculus to work through the proofs of all the theorems I have encountered along the way. What strikes me about Widder's approach is the simplicity which algebraic techniques have offered to the methods of Advanced Calculus in recent years, for Widder employs rather outdated, yet intriguing notations and techniques. In elementary mathematics, we are introduced to the four common operations of addition, subtraction, multiplication, and division. In Algebra, we are made aware that those operations come in pairs of inverses. It became clear to me that Calculus introduces us to only two more: differentiation and integration. And, it happens, that they are inverses! What separates the operations of Calculus from the basic ones, however, is their theoretical foundation: the limit. While the limit concept can be quite subtle and difficult to intuit, if we are well acquainted with some advanced algebraic manipulations, the whole of Calculus is made rather simple despite the logical quagmire of limits, and can proceed without daring to wade into it. However, attempting to prove the theorems of Calculus is incredibly rewarding, if one has t

Geometry

I have a B.S. in Theoretical Mathematics, which included the study of advanced Euclidean geometry and non-Euclidean geometry. I have five years of experience teaching students geometry. I find geometry to be one of the most important subjects in the academic career of most people. There are many reasons I could offer to sustain that point, but the encounter with logic stands above them all. Sadly, unless one takes a decent geometry course, one may never acquire any competence with logic. If I were otherwise inclined, I could easily have graduated college without ever having studied logic beyond that which I was treated to by my geometry teacher. For that matter, I could have graduated high school having avoided geometry altogether. The point being made is that too many folks reach adulthood with little to no guidance from an adequate grasp of logic, many geometry teachers included. This must be remedied. Though that needed to be addressed, I digress. Geometry is one of the oldest sciences, predated only by astronomy. It begins systematically with the cultures that arose in ancient Iraq, in which it developed as both an aid to construction and an elite subculture of visually pleasing patterns. It was the scholars of Ancient Greece, most notably Euclid, that transformed geometry from a predominantly empirical science to one built upon pure deduction. I prefer to teach students of geometry with that perspective in mind. Beginning with the assumption of complete ignorance of geometry, I challenge my students to explain why the area of any triangle is given by half the product of its base and respective altitude. It has been my experience that very few students will have a ready reply. Once students have availed themselves of justifying that formula, a matter which may take a few days, they are primed to discover all the other area formulas almost as an afterthought.

Linear Algebra

Linear algebra was required of me to earn my B.S. in Theoretical Mathematics. I went on to complete two years of postgraduate work toward earning a master's of arts in teaching mathematics, during which I took a course in the applications of linear algebra to visual effects programming. Having gained a strong interest in the subject, I continue to read and work through advanced topics within linear algebra. Linear algebra has its roots in ancient investigations into the solution of systems of two and three equations in as many unknowns. However, it has developed into a subject with far ranging applications.

Logic

One of the subjects I studied to complete my degree in mathematics is Mathematical Logic. Before I took this course, I was skilled in higher mathematics, but not masterful. It was a revelation! Not only did it make me a better mathematician, it made me a better writer. I began to see that mathematical proofs need not be frightening to the non-mathematical, to those more interested and skilled in the humanities, because mathematical proofs are nothing more than essays. They have introductory paragraphs, thesis statements, two or more paragraphs to support the thesis, and a concluding paragraph, just as the standard 5-paragraph essay model requires. Logic concerns itself with valid deduction and fallacy. There are two basic laws of deduction: modus ponens and modus tollens. The first is often referred to as the law of separation. It is best explained using the syllogism, which is a form of argument in which a conclusion is drawn from two premises. So, the law of separation goes as follows. Let P and Q be statements. Premise 1: If P, then Q. Premise 2: P Conclusion: Q The logic itself, such things as decomposing compound statements into atomic statements in order to calculate their truth values, tautologies for the conditional statement, conjugates of conditional statements, and proof by contradiction have proved indispensable in my mathematical work as well as in analyzing rhetorical statements encountered in writing, politics, and colloquial conversation. In particular, understanding what constitutes a valid argument gives me the confidence to know when I am making progress in writing the proof of a theorem. I always strive to give my clients some instruction in mathematical logic that they may gain the same confidence in reading and writing mathematics as I have gained from it.

Prealgebra

I taught students of Pre-Algebra for four years. During that time, I realized that much of the content is inductive in nature. By observing patterns, students can be led to construct the formulas and concepts with which they are required to gain facility. What better way is there to master mathematics than to discover it for yourself? I call my method guided discovery because I employ inductive processes that are naturally present in the human mind from birth to engage students with patterns that lead to constructions of knowledge, in this case, those that are required for success in future courses such as Algebra, Geometry, and the sciences. Scientific notation is a perfect concept to approach with induction. In my initial attempts to present the concept, I found that too many students had very little understanding of the effect multiplying by positive integer powers of ten has on the separatrix of a number presented in decimal, let alone negative integer powers. So, I began with an activity that asked my students to multiply a single decimal number by successive powers of ten until they were able to formulate a general statement of the observed effects. Once accomplished, it was followed with an activity that reversed the process until they arrived at the problem of intuiting what effect raising a number to the zero power, and then negative powers should have on its decimal form. At each stage in a series of such activities, I asked my students to formulate their observations into a general statement, and then confer as a class on the accuracy of the statement until all could agree. In this way, my students gained a far more profound understanding of scientific notation than I had observed in many others that were only told to move the decimal this way or that so many times as the exponent suggests. Additionally, they gained a much deeper understanding of decimal numeration, which can then be exploited to discover advanced results that broach the subject of number theory.

Precalculus

I wanted to take AP Calculus as a senior in high school, but was scheduled for Pre-Calculus instead. So, I made a deal with my Algebra 2 teacher: I would take the Pre-Calculus book home with me for the summer and teach myself. She agreed that if, on the first day of my senior year, I passed her Pre-Calculus final exam with a B or better, then I would be placed in AP Calculus the next day. I ate, slept, and breathed Pre-Calculus that summer, to the chagrin of my friends! On the first day of school, I scored an 89% on her final and the next day I was introduced to the AP Calculus class. Everyone was stunned! In Algebra 2, I figured out that I did not have to memorize endless formulas, such as sin^2(x) + cos^2(x) =1, tan^2(x) + 1 = sec^2(x), and 1 + cot^2(x) = csc^2(x) = 1, for the last two can be quickly derived from the first by dividing both sides by either sin^2(x) or cos^2(x). In Pre-Calculus, there is an overwhelming number of formulas to memorize, much of them involving trigonometric formulas more complicated than those I mentioned. However, many of them can be derived from some simple first principles. It was then I realized that logic was the key to understanding mathematical statements, and if one has this key, hours of mindless memorization may be avoided. Later, I learned that Euclid was the first to demonstrate definitively that all of mathematics rests on simple first principles. He has fascinated me ever since!

Trigonometry

While I never took a course called trigonometry, as one was never offered, there was not one math course required for the completion of my B.S. in Theoretical Mathematics that did not require the application of advanced trigonometry.