In the spring, I wrote about one of our favorite filmmakers, the Japanese animation master Hayao Miyazaki, who was known, over the span of a long and brilliant career, for creating fierce, capable, and imaginative protagonists — who just happened to be girls.

Since we happen to have a fierce, capable, and imaginative girl of our own, we’ve sought out Mr. Miyazaki’s films wherever they could be found. Often this has meant ordering an expensive DVD from an obscure warehouse, since English language versions can be quite rare.

Another specialist in heroic young ladies whose movies we’ve sought out over the years is the French director Luc Besson. For all Mr. Besson’s success in Hollywood — he was the creator of The Professional and The Fifth Element, which were both critical and box office hits — some of his films still aren’t available in translation, and a few, like The Extraordinary Adventures of Adèle Blanc-Sec, have only recently made the leap across the Atlantic.

The U.S. premiere of a new Besson movie is an event in our household, so it was with no small excitement that we plopped down in front of a big screen last week for Lucy, a star vehicle for Scarlett Johansson that has proven to be one of the summer’s big hits.

We already knew the premise of the film: Ms. Johansson’s character, Lucy, a hard-partying expat living in Taiwan, is forced into being a mule for a new illicit drug. The pouch of drugs, which is sewn into her body, starts to leak, releasing the drug into her bloodstream and endowing her with the ability to use the “90%” of her brain that lies dormant in ordinary human beings. At that point, reliable Besson mayhem ensues: the pretty girl is transformed from victim to victimizer with spectacular results.

Our impressions of the film weren’t very positive, I’m afraid. There were a few moments of brilliance, but mostly it seemed half-baked and incredibly pompous. But I’d like to set that aside for now and address the popular misconception at its heart. Contrary to what most people think, we human beings use a lot more than 10% of our brains!

The 10% fallacy is a byproduct of the early days of neuroscience in the 19th century. At the time, methods for observing brain activity were very crude, and there were large regions of the brain that seemed to simply lie dormant. These regions included the frontal lobe and the cerebrum, which today are recognized as the centers of much of our higher functioning. William James, one of the most famous philosophers at the turn of the 20th century, took his colleagues’ findings about the “dormant” brain to be a metaphor for the vast, unrealized potential of the human mind. The “10%” idea struck a chord with the public, and has survived for more than a century in the popular imagination.

The truth about the brain is a little more complicated. For an excellent explanation of how much of our brains we actually use, I can recommend a five minute “TedEd” mini-lecture, which can be found on the Web by searching for “TedEd” and “What percentage of your brain do you use?”

The bottom line is that we use virtually all of our brain, just not all at the same time. For an organ that is just 3% of our body mass, the brain uses a hugely disproportionate amount of energy — almost 20% of the total amount we produce! Most of that energy goes into keeping the entire organ in a state of electrical readiness, so that the neurons in any region can fire. But for reasons of “processing speed” and energy economy, only a select subset of neurons fire at any given time.

So it’s true that our brains aren’t 100% active all of the time. And there’s no doubt that there’s plenty of unused mental potential waiting to be unlocked in every one of us.

But films like Lucy, or 2011’s Limitless, whose plot was driven by a virtually identical premise, offer false hope for anyone who wishes that making human beings a lot smarter were as simple as synthesizing the right drug.

Lucy and Limitless are interesting films to compare. They both ask the question, “What would you do if your mind were suddenly completely awake?” Limitless, which was directed by an American, offers a protagonist who starts out as a writer, but uses his newfound powers mainly to make money, which proves to be a springboard to political power. Lucy, which was directed by a Frenchman, starts with a lost soul who at first seeks vengeance, and then justice, before turning her infinite mental powers to the quest for pure knowledge.

I like the French version better, although in both movies, as soon as the protagonists get super-smart, they immediately shed the pesky limitation of human ethics. This is Nietzsche’s idea of the will to power: i.e., the strong ought to be able to do whatever they like, since, after all, they are superior.

I much prefer the supreme being in Mr. Besson’s masterpiece, The Fifth Element: Leeloo, who uses the perfection of her mind and body to protect the weak from the overwhelming forces of evil.

Now that’s a message a parent can get behind.