Some of the best mathematicians that I have had as students did very poorly in elementary school math because they were focused on the why question and were never satisfied with the answer. “Why does subtraction require borrowing?” “Why do you flip a fraction and multiply when you’re told to divide it?” “Why does long division even work?” In fact, so many students of mine were prevented from advancing and given damagingly negative feedback when they were faced with long division and just never got the process that they gave up on success in math for a long time. Math became a mysterious subject where the teacher got it and other students seemed to get it, and they just felt lost.
The fact is that getting long division problems right is about following (and memorizing) an algorithm, or set of instructions. Understanding why the long division algorithm works is quite challenging, and actually far more advanced mathematics than is generally found in most high school curriculae. I demonstrate it using polynomial long division in my Trig/Pre-Calculus classes, but the real proof was something that I came across when studying Number Theory, an advanced math class I took in college, and continued to pursue in my career as a mathematician.
The truth is that numeracy is far more important – understanding what division really means is much more meaningful than memorizing a seemingly random set of instructions that involves lots of guessing and checking with larger numbers. I often tell my students that the next time they are likely to use long division after high school will be in (hopefully) at least 15 or 20 years later, when they have their own kids in elementary school. And maybe, by then, that algorithm will no longer be the barrier to higher math that it is now, and has been for far too long.
What are your thoughts on long division? Is it really useful? Why?