Knowing how to drive a car and fixing a car are very different. Using a computer and fixing something wrong on the computer are very different skills. What do you need to know to drive a car or use a computer? What decisions do you need to make very day?
In mathematics, students have learned to compute by specific sets of rules for years. The part that's been difficult for some of these children has been when they have to make decisions on what procedures to use and when to apply them in any other settings other than isolated computation. They often made errors using the procedures which led them to incorrect answers and they didn't even realize it because they lacked number sense.
Children do need to know how the basic operations work to really have the knowledge to use them meaningfully. They need to develop a variety of operational procedures to solve a problem. It's important to realize the relationships between operations. They need to be sense makers to make decisions about how to solve the problems. That requires more than knowledge of procedures.
I've only addressed number operations so far. I guess because it seems to be the topic most associated with mathematics. However, I think the same for all areas of mathematics and learning. Students need a variety of spatial experiences to understand geometry ideas. They need experiences with data to understand how it's collected and interpreted. Many of the ideas explored in these experiences then link to other mathematical experiences. It's not that all children are being trained to be formal mathematicians. We all need these skills in real life. We use logical reasoning skills developed through mathematics every day when we try to solve problems. To know means to think and reason.
I think we use sense making when we use our cars and computers too. We make connections and learn more with our many varied experiences. Our knowledge base grows through these experiences so we are not just following a set of procedures. We are generalizing and applying knowledge we have acquired through a variety of experiences. Computers and cars work no better than the operator using them. If the operator can think and reason, then they have many more options in the ways they can drive a car (drive in different settings like the city, or when it snows; drive cars with automatic or standard transmissions; become a bus driver or truck driver) or use a variety a computer can provide (use a word processor; send email; search the internet; build a database; developing web sites; set up a mailing list). Learning never stops, even when it comes to cars and computers. I think what makes learning engaging and some times even fun is learning how things work.
I think the following articles address your questions about mathematical learning:
