Mathematics is definitional, conceptual, and relational.
The Overall Goal
This is a tricky subject to teach. It’s easy to simply have students memorize lists of formulas or teach them to work specific kinds of problems. However, this is neither the heart nor the depth behind mathematics. I want my students to learn the definitional (which is necessary for discussing ideas), the conceptual (that is, core concepts behind what we learn), and the relational (that is, how definitions and concepts come together to create mathematics itself).
Ultimately, I teach the essentials of what they need and then work with them to figure out remaining concepts on their own. For example – we don’t just memorize the formula for the area of a circle. We take a circle, cut it up into pieces, and work with it until the student can create that formula all on their own. It is an unbelievably more effective approach, and it creates a student who can tackle problems on their own instead of needing answers given at every step.
What I’ve found in teaching mathematics (and every topic, really) is that students enjoy learning to a much greater extent when they feel a sense of discovery. Memorizing formulas is not discovery, it’s just repetition. So we go deep and we work hard. We laugh, we make jokes, we make fun of our mistakes. And, when it’s all over, my students know that they have learned not just mathematics but a character that isn’t afraid of hard work and careful thinking.