My goal as a teacher is to inspire students to learn and to grow, both within my subject and as people in general. Each student who leaves my classroom with a greater maturity and a better passion for the field of study is a reflection of that success.
Mathematics education in this country has been persistently problematic. On the one end of the philosophical spectrum is the view that mathematics education should involve teaching students a stream of theorems and algorithms, with a heavy emphasis on teaching specific concepts that might appear on standardized exams. On the other end is the view that students should learn a mindset of how to approach mathematical problems, with the implication that students will be able to apply those tools to any problem they might encounter.
I believe the most effective way of teaching mathematics is a mix of these extremes. There are certain pieces of information that must be learned, but at the same time, it is important to make connections and understand the logic behind the algorithms.
As a geometry teacher, one example I use is the Pythagorean Theorem and the Distance Formula. While proofs of the Pythagorean Theorem exist in multitude and those proofs are useful for understanding the theorem, the theorem itself is nonetheless something that should be committed to memory. On the other hand, there are two ways to access the Distance Formula. Students who are geared towards rote memorization will tend to simply memorize it; students who are geared towards understanding will come to see it as an application of the Pythagorean Theorem.
As a secondary teacher, I believe a significant part of my role is to guide students from the rote memorization of concepts that they may have become used to in the elementary school years towards a deeper understanding of connections, while at the same time pointing out those elements of procedural fluency that simply must be memorized.
Mathematics is a complex field to teach because of this tension between procedural fluency and the strategic and adaptive reasoning expected of competent mathematicians. It is also complex because of the tension between the practical and the abstract.
In elementary years, there is emphasis on the practical: Operations on real numbers and ratios are things that can be easily shown. Students can see how "My friend has three apples and gives me one" might apply to their personal life. In secondary school, there is a transition towards the abstract; students struggle to see how these more complex issues apply to them. There is a drive in mathematics education to see how students can apply these abstracts to their own lives, through project-based learning.
While I support project-based learning, I also fear that some of the beauty of mathematics as an abstract art gets lost in the drive to relate everything to a response to, "How am I going to use this in real life?" ELA teachers manage to communicate the intrinsic beauty of their field: The joy of literature in its own right, as well as its application to the reader's world. Why can't mathematicians do the same thing?
For example, consider two chords intersecting inside a circle, forming four segments. If you multiply the lengths of the two segments on one chord, you will get the same product as if you multiply the lengths of the segments on the other chord. There are rarified practical applications for such a fact, but for the most part, I think the core reason for discussing this characteristic is because it's just plain cool. Imagine: Wherever you put a point with regards to a circle, then draw a secant line (it also works outside the circle!), the product of the lengths of distances between the point and each intersection with the circle will be constant for that point. That's nifty. That's art.
So my goal as a teacher of mathematics is three-fold: (1) To teach basic facts about mathematics that will be useful to students in their lives; (2) To guide students from a "rote memorization" perspective to a habit of reason and problem-solving; (3) To inspire students to see the beauty in mathematics. Effective teaching involves guiding students towards a deeper understanding and passion for the teacher's field of study.