When I was in elementary, learning the basics of arithmetic was an important component of my mathematics education. I participated in various mathematics “challenges”, where I tended to do pretty good. I like the rush of having to beat someone to an answer in a competitive setting, so I became good at it.
Fast forward a few years (and even to today), and people seem to be astonished when I crank out answers to arithmetic faster than they can input the numbers into their calculator. Honestly, I’m not even that fast or that good, but knowing some simple patterns in counting allows me to appear as if I have super powers.
What I find is so interesting (and unfortunate), is the reaction that these people have. Once they see that I can compute quickly, they tend to say, “Wow, you’re pretty good at math!” This is a nice complement, but they miss the point of what it means to be good in mathematics. On the one hand, speed is important. After all, if you considered two people answering a multiplication question, I doubt you would say that the person who answers the question correctly, but slower, is the better person. You probably wouldn’t say that they have the same ability either, because the first person answered quicker.
This is true for brute mathematical calculations, and those become the work of computers. The faster, the better. However, the idea is that what you will focus on is answering questions that are much more deep and complex. You will need to develop your problem solving skills to figure out the answers to questions, something a computer won’t be able to do for you (at least, for now). You may not be faster than another person, but it doesn’t matter because you are tackling brand new questions that nearly nobody else thinks about. Yes, speed is important, but an awareness of the strategies needed to solve the problem are just as crucial.
In essence, being good at mathematics isn’t about crunching numbers quickly, it’s about knowing the process and being able to “spot” what tools are needed for the problem at hand.
Unfortunately, the present situation is that we call those who can quickly compute numbers early on as mathematical “geniuses” who are just so smart. What this does is encourages the ones who are doing well (which isn’t a bad thing), but discourages those who can’t calculate quickly yet (which is a bad thing). This distinction makes it possible to push students away from mathematics because they don’t feel they have the right “stuff” from a young age.
Instead, we need put forth the message that calculating quickly is great, but it’s much better to get the process right first. It’s better to get the right answer in a long time than the wrong answer in a short time.
Basically, speed matters in mathematics, but only for the sake of increasing productivity. While learning, it’s more important to teach the process and not alienate students because they aren’t deemed “smart” since they can’t speed through mental arithmetic. That’s not what mathematics is about.