By Adam J. Pearson
Many math teachers fail to grasp the importance of ‘why’ questions from their students or simply choose not to engage them when they come up in their classes. For many students, this choice causes a great deal of frustration, confusion, and exasperation. Indeed, I very nearly almost failed introduction to linear algebra and pre-calculus because whenever I would ask “why do we do this step?” my teacher would answer “Because that’s what you do.” I know I was not alone in this experience. This “answer” is not in fact an answer at all. It does not address the question; it only dismisses it. Such an answer does not increase understanding; it only increases frustration and fear and hatred of the the beautiful and vast subject that is mathematics. And that is precisely the opposite reaction to what we aim for as teachers: love and understanding of the material.
As a math teacher, I almost always answered “why” questions. The only time I didn’t was when the answer would require me to teach multiple other higher level concepts that the student wasn’t ready to grasp. In most cases, though, I did answer those questions. Why? Because they are extremely intelligent questions. In daily life, we don’t do things without knowing what those actions contribute to our overall goal. We know that if we are sweeping the floor at McDonald’s, then we’re doing that both to keep the floor clean and to ultimately get paid. We know that if we put draino into a drain, it’s not for no reason, but to help unclog the pipe. So why should math be any different?
Moreover, understanding the ‘why’ of each step we do in math helps us to see the big picture of the whole problem in a clearer light. This is just like how it helps to understand what each ingredient contributes to a recipe if we want to get a feel for how the recipe ‘works;’ understanding how this ingredient adds saltiness, this one sweetnesss, this one spice, etc is key to seeing a meal through the eyes of a chef. In the same way, it helps to understand how the gears in a machine work if we want to know how the machine as a whole functions. Similarly, it helps to understand the ‘mechanisms’ that make our math tools work. When we know the nature of our tools, we can figure out how to use those tools to solve problems.
This precise understanding of the ‘whys’ and the details of math is what true mathematical understanding involves. A student who has simply learned to imitate the ‘correct steps’ that the teacher showed him in a rote manner has not truly understood the math. They’ve been shown the tip of the iceberg, but denied the great depths below the surface of the water. Seeing how and why your math tools help you to arrive at a solution is the mark of true understanding in math. And that’s why asking and answering “why” questions about the steps and operations we use in math is so important; such questions and answers are the keys that help unlock the doors of deeper understanding.