Methods or Principles?

When it comes to solving a math problem, there are several methods a student can implement that will yield the correct answer. Sometimes, while helping friends with their math problems, I like to wait until they finish the problem to confirm their answer matches mine. This approach gives allowance to the fact that we learned two different methods for solving that sort of problem. However, because I grasp the general principle that they are applying, I can often double-check their work even if the specific notation or order of steps is different than what I utilized.

I have noticed that my understanding of general principles is not shared by all math students. This makes me question whether math teachers are teaching methods or principles to their students in the classroom. Many students struggle to solve a problem unless they are using the specific method they learned in the past because they do not understand the underlying principle behind the method.

“As to methods, there may be a million and then some, but principles are few. The man who grasps principles can successfully select his own methods. The man who tries methods, ignoring principles, is sure to have trouble.” -Harrington Emerson

Although this quote is usually applied to life principles, I find that it is just as relatable for a mathematics classroom, while also addressing the very observations I have made. Math teachers need to be teaching their students the principles and big ideas of a lesson. If a student understands what they are trying to accomplish and the principles behind finding the solution, they can then choose the method for solving that makes the most sense in their mind. Students are often taught multiple methods to solve a problem, but without an understanding of the foundational principles of the lesson, they will struggle to know which methods to use, why they are using that particular method, and when to use it.

A competent teacher helps their students understand the principles of math so that they can choose a method that makes sense to them. Teachers serve their students well when they expose their students to one main principle and multiple methods that address that principle. This idea is utilized in both lower and higher-level math classes. In Calc 2, I learned many methods for solving integrals. However, I first needed to grasp the principle of integration if I was going to be successful in choosing which method to apply to a problem.

Gaining a deep understanding of mathematical principles also means that students will be able to solve many different problems. If all a student knows is a method for solving a given problem, they will be confused when they run across an unfamiliar problem even if it uses the same basic principle. However, if the student has been taught the mathematical principle, they will have the foundational knowledge needed to solve problems that initially look unfamiliar. Methods are often easier to teach, but teaching principles are what will serve students best as they move forward in the world of mathematics. Teaching principles allows students to become problem solvers, not merely method masters.