One of Columbia Business School’s most popular courses in recent years has been in strategy, called Napoleon’s Glance, named after a book by the instructor, Bill Duggan. Former students I’ve talked to rave about it.
I was fortunate to do an independent study with him before his course exploded in popularity. Now it’s so successful I doubt he could devote that kind of attention to a single student.
Despite the course’s immense popularity at one of the world’s great business schools, I was more influenced by his book The Art of What Works. This excerpt from the back cover gets the gist of the book well for me, especially Jack Welch’s quote:

While every business situation you confront is unique, it is invariably made of questions and elements that have been confronted–and solved–by others before you. The Art of What Works outlines a step-by-step program for understanding how and why others succeeded, and then drawing on their successes to help solve your own business problems. Outlining an approach that is exceedingly straightforward yet dramatically effective, this landmark book will help you to:

• Systematically draw on the past successes of others to fuel innovations of your own
• Lead effectively by learning how to construct one, dramatic solution from several elements
• Overcome obstacles that prevent good ideas from taking shape, and rising to the top

What has worked in the past, more often than not, will work again in a new combination. The Art of What Works reveals how to transform this intuitive observation into a structured program designed to save time, energy, and money for both yourself and your organization, by giving you the freedom to recognize–and rely on–the simplicity of what works.

“The operative assumption is that someone, somewhere, has a better idea; and the operative compulsion is to find out who has that better idea, learn it, and put it into action–fast.”

–Jack Welch

This book changed my problem solving starting point from trying to isolate myself from the world when solving problems to starting with the question “What problem like this has been solved before and how can I use that solution here?” This perspective leads to faster solutions more likely to succeed and, frankly, more fun to do.
Bill calls this technique the art of what works. Obviously there’s more to it than my brief description. The book goes into a lot more depth, describing how others put it into practice successfully throughout history so you can put it in practice in your professional and personal life.
I recommend thinking this way too.

What’s valuable in solving problems

This perspective, along with plenty of independent research, suggests some valuable things in problem solving are

• experience solving other problems in relevant areas
• knowledge in the field

As someone who spent many years developing abstract problem solving ability — math and physics tend toward abstract problems — I was surprised how little abstract problem solving helped solve non-abstract problems. (I should cite sources. I’m pretty sure Weisberg’s Creativity: Beyond the Myth of Genius, listed on my resources page, covers it)

Why companies pay more for experience

Nearly everyone pays more for experienced people than inexperienced. They pay because experienced people have the two things listed above (assuming you interview them well) — the most valuable things to solve problems, and what do you hire people more for than to solve problems?
People have experience solving problems in relevant areas and they know the field can draw on and apply past solutions faster and more easily than others — in other words, they can solve problems better.