There is a divide between pure and applied math, one that arguable should be bridged, but the divide you talk about /within/ applied math seems artificial: Either you learn math properly, at least with regard to depth as opposed to breadth, or you shouldn't be working with it in the first place.
Within electrical engineering the design process is 1. pen/paper + math 2. simulation 3. building/testing. You learn quickly that jumping straight to 3, or even 2, is a /bad/ idea. This is because in order to isolate where the problem lies, you need to know what is expected. A similar process is used within any engineering discipline.
There is a such a thing as software engineering, where in the interest of safety the software system is documented completely. But in the vast majority of case, I hold the outre opinion that programming cannot in good conscience be considered engineering.
In conclusion: If you are working in engineering or applying math, you should learn math properly, and shouldn't treat it as a black box.
Within electrical engineering the design process is 1. pen/paper + math 2. simulation 3. building/testing. You learn quickly that jumping straight to 3, or even 2, is a /bad/ idea. This is because in order to isolate where the problem lies, you need to know what is expected. A similar process is used within any engineering discipline.
There is a such a thing as software engineering, where in the interest of safety the software system is documented completely. But in the vast majority of case, I hold the outre opinion that programming cannot in good conscience be considered engineering.
In conclusion: If you are working in engineering or applying math, you should learn math properly, and shouldn't treat it as a black box.