The place of pure mathematics is changing...
Here's an excellent description of what is supposed to happen when you need to solve a problem that requires math:
"Here's how to exploit mathematics to do applied problem solving. In a transportation analogy, we start with a real problem a point A and we want to get to a real solution at point D. For this trip we can start out walking, without a map, across deserts, rivers, swamps, and oceans, and maybe get to D. But here's what we SHOULD do: First get a taxi ride to a local airport Assumptions at B. At B get a plane trip on Mathematics Airline to another airport Conclusions at C and close to D. Then take a taxi to D." -- HilbertSpace
Former: how computer vision is developed. Latter: how Pixcavator was developed. It was an exercise in fairly simple mathematics that turned out to be very practical. The variety of fields where it has been applied is amazing!
Meanwhile, the lack of understanding of mathematics in the so-called "technical" fields is appalling. See especially Bad math.
See also Patents and mathematics.