I attended Texas State University, where I earned two Bachelor's of Science, one in Mathematics and the other in Physics, and I continued on to earn my Ph.D. in Mathematics Education. I have been tutoring students in mathematics and physics since 2008, so I have been learning about and refining my teaching methods for over a decade.
I view mathematics as a great, inverted pyramid, wherein each new idea is built upon a foundation of interwoven concepts. However, throughout my twelve years of experience working with students, I have found that many students seem to believe that mathematics is merely an ever growing list of discrete facts and that learning mathematics is a matter of just memorizing these facts in preparation to regurgitate information for exams. In my view, students who have been exposed to this disconnected view of mathematics are often overwhelmed by the sheer volume of information and seek to memorize only the particular steps and equations necessary to solve each type of problem, rather than trying to understand the overarching structure and reasoning that can be applied in a wide variety of contexts to solve many different problems. Therefore, I deliberately and explicitly focus my teaching on: connecting mathematical ideas, and modeling reasoning about structure.
Connections: I explicitly connect the current topic to prior mathematical content. For example, during integral calculus courses when discussing integration through trigonometric substitution, textbooks often present a table showing three categories of algebraic expressions along with their corresponding trigonometric substitution and reference triangle. Rather than show these to my students and encourage them to memorize these particular examples, I instead prefer to show how we can use the Pythagorean Theorem to build a right triangle based on a given algebraic expression and then, using knowledge of the trigonometric functions, how to find a useful trigonometric substitution. I find this to be a more useful way to approach these problems because this method allows my students to apply the same reasoning to all integration by trigonometric substitution problems, thus affording students greater opportunity to be successful when they are working on their own. Moreover, this approach can help students avoid common memorization pitfalls, such as using a trigonometric substitution and a reference triangle that do not match.
Modeling Reasoning about Structure: To help my students develop problem solving and reasoning skills, I provide a model of appropriate mathematical reasoning by thinking aloud as I work through problems, discussing things such as which features of the problem stand out and indicate possible solution methods. In order to help my students try to imitate my cognitive processes, I organize my problem solving methods around a series of questions that I ask myself about the problem in order to determine how to progress. In the example of working on integration by trigonometric substitutions, I first discuss how each other method of integration (e.g., substitution, by-parts) would be inappropriate, and then examine which form of the Pythagorean Theorem resembles the algebraic expression in the problem. I then use this information to build a right triangle that describes that form, and finally use this triangle to choose an appropriate substitution. As my students work, rather than tell them the next steps, I instead first refer back to these organizing questions and try to get them to find answers themselves. I prefer this approach because I believe, and explicitly discuss with my students, that understanding the rationale underlying the steps in a process is better than merely memorizing the steps for two reasons: the rationale's structure reflects and reinforces the structure of the content, and using a process consistent across examples reduces the amount of information that students need to know in order to be successful on assessments.
I have also worked with future teachers as an instructor of a first course in foundational mathematics content. In this context, it is of paramount importance that my students understand why the mathematics works like it does and how the various topics are interconnected so that they are able to provide instruction to their future elementary and middle school students in ways that emphasizes the connections among various models, between models and algorithms, and between prior and new content. For example, while discussing the base-ten number system, we explore how manipulatives (e.g., toothpicks, beads) can be iteratively bundled into groups of ten to demonstrate how place value is used and that this process can continue forever. Then, with non-integer real numbers we capitalize on this pattern to justify the idea that we can, similarly, divide a single unit into 10 smaller pieces (i.e. into 10 tenths) and that this process also continues indefinitely. This understanding of bundling and unbundling groups serves to emphasize the consistency of the base-ten structure across all place values, and also provides a basis for the "carrying" and "borrowing" actions present in the standard algorithms for addition and subtraction. By showing these future teachers connections among mathematics, and by modeling reasoning about structure, I hope to affect change in their views of mathematics as a discipline as well as provide an example of how we, as teachers, can help guide our students by relying on the structure of mathematics and through responding to student questions with carefully chosen questions of our own that encourage students to think for themselves.