When working with students I often get asked the question, “How do you know all of this?” or “How do you do math so well?”. I feel the answer to these and similar questions is just as simple as it is for anyone learning to do anything. When you study something thoroughly enough, you begin to piece everything together as one whole truth, if you will. In other words, when we focus on math as one whole truth and refrain from getting caught up or lost in the steps and procedures and forget what we learned, we begin to understand how each lesson (or piece of truth) is preparatory for the next. Think of it as a cake with infinite layers and each layer necessary before the next. If a layer is missed, then the next layer will not fit quite right.
I am not sure why, but I notice far too often, that students “learn” math, take a test, and then forget it. When you show them or discuss a concept that they already “learned”, they often act like they have no clue what you are talking about. At some point, they must approach math as a whole subject and put the lessons together.
For example, think of a knitting a sweater. One might knit an arm, front, back, etc. At some point, in order for the end result to actually be a sweater, it must be all pieced together.
Or, one can compare it to learning to drive. A new driver might study operating car controls, learning laws and road signs, parking techniques, interstate travel, backroads, night-time driving, etc. Any one of these lessons standing alone will not create a skilled driver. Nor will doing any of these things just once or twice. It’s in the actual piecing together of the skills and practice of driving in each area, that one becomes a skilled driver.
So it is with mathematics. To my elementary students, learn what you are doing now very well, because you will never stop using it in the progression of your mathematics study. Middle school students, what you are learning now is preparatory for high school mathematics. High school students, by now I hope that you are starting to piece together this beautiful subject of truth called mathematics! What you are learning is a culmination of all of your earlier years and preparatory for you to advance in further mathematics. There is always more to discover and learn and the further we go, we begin to see the subject as one whole truth. Our perspective and understanding is enhanced.
Thus, I would suggest as a final answer to these opening questions is that I have learned to understand math and not just memorize it. So, I approach it by applying all truth that I know and putting the question in context, then solving. I have spent much time doing and practicing and that is “how I know all of this”! Anyone else can, too.