Here is the thing. Math is supremely abstract as it is taught. You go from “you have two apples and Betty has 3 apples, how many apples do you have together?” converted to 2+3 and then you are in the abstract math land until you graduate.
To elaborate. Math is a language that helps solve problems. Math education is there to help the student abstract from real world to math language to solve problems using the patterns available in the language. The goal should not be to be able to do calculations without resorting to using calculators but rather being able to take a real world problem and abstracting it into math language.
The problem is in the way we teach. To solve a problem in real life we need to follow the progression: 1. problem -> 2. abstraction -> 3. calculation -> 4. solution. The way we are taught now is by giving us the predigested problem. We are given the the problem in the already abstracted form. “How much is 3+2?” is the abstracted form. We are expected to do the calculation in this chain, the third step to reach the forth, and the result is checked for truthiness to grade how well the calculation is done.
Most people can follow the abstraction to multiplication, but when you come against problems that require logarithms for instance they are at a complete loss. This is not a problem of the algorithm being to complicated to be applied to the problems at hand as you suggest but rather the inability to abstract the problem to mathematical notation.
You claim that the problem is that we are taught to use algorithms that work on paper when we should be taught algorithms that work purely in our mind. Your solution falls into the exact same trap that the current math teaching does — the calculation is trivial.
We have algorithms for solving problems on paper, for solving them in our head and also ones for solving them in the silicon brains of our computers. They are all different but the latter do this at a fraciton of the time and cost. When a student asks “When do we use this in real life?” they really ask “If i have a calculator then why would i need to use this algorithm designed to be used for doing calculations on paper?”. The answer is that you really dont need the algorithm atall and your teacher is wasting your time teaching it but too blind to see it.
The reality is, that i as a person who applies mathematics at a relatively high level as my day to day job, i dont do any of it on paper nor in my head. What i use is in this case is the language R. What i need is the ability to abstract from a real world problem to the mathematical notation that i can feed into a computer to get the solution. The algorithm is irrelevant as computers do it differently anyway and unless you are going deep with computer science or mathematics then you should not care about it.
The calculation algorithm does not matter. Its all about abstraction.