As a mathematician who made the jump to software I feel that, while there is certainly room for improvement in math education, methods based too much on intuition and feel are a hugh step backwards.
Math is about thinking abstractly. To do it at the level required by modern science, data analysis and engineering you need to be able to focus on the abstract symbols, equations and rules that govern them without relying on an intuition for underling objects. For example, no one has valid intuition for fluid turbulence, n dimensional manifold theory or complicated probability distributions, so great leeps in understanding these are made by people who have an intuition for how equations behave and rigorously show that it is valid. Real world applications are often only found after the fact.
I think that they key to a good math education is not just showing students real world application but teaching them to find beauty and pleasure in abstract symbolic reasoning and the rigors of proof.
Math is about thinking abstractly. To do it at the level required by modern science, data analysis and engineering you need to be able to focus on the abstract symbols, equations and rules that govern them without relying on an intuition for underling objects. For example, no one has valid intuition for fluid turbulence, n dimensional manifold theory or complicated probability distributions, so great leeps in understanding these are made by people who have an intuition for how equations behave and rigorously show that it is valid. Real world applications are often only found after the fact.
I think that they key to a good math education is not just showing students real world application but teaching them to find beauty and pleasure in abstract symbolic reasoning and the rigors of proof.