Everywhere you turn, you hear of the demise of the U.S. education system. ExxonMobil recently ran television ads that say the U.S. is now 25th in math and 17th in science worldwide. Nearly every study shows that U.S. students are falling behind countries such as India, China, and South Korea in education, yet American companies seem to be holding their own against competition from these countries. The U.S. may not be making the world's most popular items, but we are inventing them. How could this be?

One major factor that these studies don't show is what the U.S. leads in: innovation. Innovation is a skill that's hard to define but easy to identify when we see it, and we see it more in the U.S. than anywhere in the world. Maybe it's a level of irrational confidence we're raised with, maybe it's the creativity that's accepted in our culture, or maybe it's harder to define than that. Whatever it is, there's something intangible that's different in a U.S. student than our international counterparts.

A Fool's two cents
I was reminded of this last week when talking to a friend who is teaching elementary school in South Korea. She said that the students are brought up to work extremely hard and value an education from an early age, often going to academies after school to prep for high school entrance exams, but have a hard time with creativity and problem-solving skills. When she gave first- and second-grade students freedom to create an Easter basket from a plethora of supplies, most students made the exact same basket as she had made as an example. In a sixth-grade class trying to teach debate skills by arguing the merits of school uniforms, students couldn't understand benefits or drawbacks that didn't directly relate to them. The focus on learning by rote had become more important that things like creativity and context.

I've experienced this in my own education. In engineering school, there was a large percentage of international students, most of whom scored better on tests than I did. But translating book smarts to the real world was a challenge for many of those students. The smarter they were, the more degrees they had, but the less likely they were able to apply their education to making products or innovations people would actually buy. The teams with the greatest likelihood of success may have included the smartest people in the room, but the team members grounded in reality were just as important to success.

These are experiences I have had, augmented by anecdotes from others, but the tangible evidence is there as well. The industries we think of as dominated by the same foreign companies that are "smarter" than we are actually rely on U.S. innovation to fuel the products they make. Here are three high-profile examples.

The U.S. stopped being the hub of computer manufacturing more than a decade ago. Today, it's more than likely that the computer or other device you're reading this on came from somewhere in Asia, not the United States. But that doesn't mean the U.S. doesn't dominate the ideas and innovations that go into those machines.

Apple (Nasdaq: AAPL), which has come under fire for its outsourcing of manufacturing to Foxconn in China, still packages every product with the key words, "Designed by Apple in California." Those five words describe our innovation advantage better than just about anything else.

Microsoft and Intel (Nasdaq: INTC) still form the heart of most PCs, and while they're global players, most of their research centers are domestically based. Intel has nine research centers in the U.S. compared to two in China. Microsoft says that "the bulk of Microsoft researchers work out of the Redmond, Washington, campus." It even mimicked Apple's tag by putting "Hello from Seattle" on the Zune.

Even Samsung, which has risen to power in smartphones and tablets, uses Google's operating system, which was developed in Palo Alto. You could even argue that the physical design of Samsung's phones came from the U.S., since a jury recently found that Samsung copied Apple's designs.

It would actually make more sense in a lot of ways to have products developed closer to where they're manufactured. But for the most part, China, Taiwan, South Korea, and other countries have yet to take the innovation torch (arguably with the exception of Samsung).

Another industry I follow, the solar industry, is dominated by Chinese manufacturers, and even U.S. firms like First Solar (Nasdaq: FSLR) and SunPower (Nasdaq: SPWR) make most of their panels in Asia. But the ideas behind solar and the technology advancing its development are made right here at their U.S. headquarters.

Bell Labs produced the first solar cells for space use and later produced the first modern silicon solar cell. Today, SunPower and First Solar are researching and advancing solar efficiency in the U.S. and taking the technology to their plants in Asia. Chinese firms are well behind the industry's lead and make modules that are nearly identical.

Advancements in solar are also being pushed by GT Advanced Technologies (Nasdaq: GTAT), a company that manufactures equipment used by Chinese firms to make polysilicon and ingots (the foundation for solar cells). This technology is being put to use in China, but again, the ideas and strategy behind it are coming from the U.S.

Foolish bottom line
Considering the fact that the U.S. isn't the most tax advantageous place to do business, our workers aren't as educated as many countries overseas, and our cost of labor is higher, there must be a reason so many of the world's best companies are based here. The best answer I can find is that we have an innovation gap over the rest of the world, and companies value that more than just cost.

Next time you see a list of the most innovative companies in the world, think about how many of them are U.S. companies. They may not manufacture their products here, they may not sell most of their goods here, but their ideas are truly American. It's not a tangible skill that can easily be tested, but I think there's clear evidence that we lead the world in innovation, and I think that's far more important than simply rating how we score in the classroom.

For more on how American companies stay ahead, check out our detailed reports on three of the companies mentioned here. Our reports highlight how Apple generates its incredible growth, how Intel plans to stay on top, and how First Solar stays ahead of Chinese rivals.

Fool contributor Travis Hoium owns shares of SunPower and manages an account that owns shares of Apple, Intel, and Microsoft. You can follow Travis on Twitter at @FlushDrawFool, check out his personal stock holdings or follow his CAPS picks at TMFFlushDraw.

The Motley Fool owns shares of Apple and Under Armour. Motley Fool newsletter services have recommended buying shares of Google, Intel, Microsoft, Apple, Nike, and Under Armour, as well as creating a synthetic covered call position in Microsoft, a diagonal call position in Nike, and a bear put spread position in Under Armour. The Motley Fool has a disclosure policy. We Fools may not all hold the same opinions, but we all believe that considering a diverse range of insights makes us better investors. Try any of our Foolish newsletter services free for 30 days.