The most common approach used in school for teaching arithmetic operations is algorithmic, where you write numbers down in columns and do pre-determined steps. A good example is this multiplication problem:
The thing about this approach is that it is not only systematic but pre-determined, and based on strict rules of operation. In effect, it doesn’t require any actual thought to carry it out; once you learn the rules you can perform them robotically, without any conscious engagement. In fact, it can be carried out most reliably by electronic circuits that ostensibly have no consciousness whatsoever, and the characteristically human traits of imagination and creativity only hinder the process. But (surprise, surprise), your brain is NOT a calculator, which is why you can only perform this sort of operation with the help of a pencil and paper. (If you don’t believe me, go ahead and test this by performing the multiplication above in your head WITHOUT writing anything down, and without looking at the picture. I bet you can’t do it.) This type of algorithm is what you would use if you wanted to program a computer to do multiplication (which is mainly what it is good for), but it is not optimized for the way the human brain works.
As an alternative, consider the following approach to performing this calculation, that actually aligns with the way your brain naturally works, and takes advantage of its powers of creativity and visualization. Without writing anything down, see if you can perform the calculation this way:
123 x 45
123 x 5 x 9
123 x 10/2 x (10-1)
1230/2 x (10-1)
615 x (10 – 1)
6150 – 615
6150 – 15 – 100 – 500 = 5535
Each step in this calculation is one you can perform easily in your head, and, I’m willing to bet, with a little practice at least, you could go through the whole process mentally in less than 10 seconds without writing any of it down.
The thing about this process is that it is NOT algorithmic. Rather than following an explicit series of steps that works in all cases, we chose one of many possible sequences of steps we could have used that required a minimal amount of mental effort. (There is, in fact, an infinite number of other sequences of steps that would arrive at the same result, some of which may be even easier.) This approach requires creativity and imagination; you couldn’t program a computer to do it. The individual steps are easy; the effort lies in visualizing a number of alternatives and choosing the best one, but fortunately this is the sort of thing that our biological neural nets, i.e. massive parallel processors, are designed to do. This approach takes advantage of our unique human capacities rather than short circuiting them. It also shows that while it is true that in math there is usually just one right answer, this does not mean that creativity and imagination don’t have a role to play, as there is always an unlimited number of ways to discover it.
Here’s an article where math professor Sanjoy Mahajan demonstrates something similar with long division.