can 5-year-olds learn calculus?

You can check out this book by Maria Droujkova here

I stumbled on this article from The Atlantic over the weekend, and I have really been giving it some thought.  Take a couple minutes if you like and read it here, or you can just read my summary below.  

Some summary points:

• The way we teach math is contrary to the way the human brain, children, or mathematics works.
• Early emphasis on arithmetic is bad for kids and can lead to negative attitudes about math.
• The complexity of an idea and the difficulty of doing it are different.  An idea can be simple to understand but difficult to do.  
• Games and free play are efficient means by which children learn.
• By exposing children to the ideas of calculus and algebra early you build a "canopy of high abstraction that does not whither."
• There are levels of understanding and there is no expectation that children will have a formal understanding of the mathematics early on.
• Children need a voice in their learning and the ability to choose what and how they learn. 
• Push back to these ideas comes from two camps. The first are those that think parents will push their kids too hard when they find out that they can "do calculus" in elementary school.  The second group thinks that essential computational skills will be lost.  

I have been teaching AP Calculus AB for more than 20 years and there is very little about this curriculum that I have not mastered at this point.  I have a clear understanding of the large questions that are answerable through the use of calculus and the details required to reason to a correct solution to such questions.  

I think that I have some real questions about the practice of leading children to calculus suggested by Maria Droujkova.  Some of it comes from the fact that for the past twenty years, my most successful calculus students were those that had completely mastered the foundational arithmetic skills required in support of the calculus.  I can certainly have a conversation about fractals and even wander into the ideas of infinitesimals with a 1st grader, but there is literally no chance that anything will come of that conversation other than, "This is cool."  Even if this child was taught some kind of procedural steps that led to a solution to a question they asked, I am sure they would not have any understanding of why the did any of the steps or what any of it meant.

One of the hardest things I have to do is teach my students how to see the forest for the trees.  Every day we look at the toolbox that is calculus and discover the amazing problem solving tools that allow us to solve both practical problems of changing quantities and growing volumes, but also to logically prove why a particular formula from geometry or technique from algebra actually works.  In many ways I feel like calculus is a culmination of a dozen years of mathematics, giving students the means to literally fly above the trees of questions and see the connections, the intertwining branches of the entire forest.  

So perhaps this really comes down to a matter of semantics.  What does it mean to teach a 5-year-old calculus?  What does it mean to teach the same subject to a high school student?  It's simply not the same thing.  And in my mind, that's ok.  Showing a 5-year-old that math is cool is absolutely fine with me.  Please, go ahead and spark curiosity!  Show kids fractals, wow them with infinity, expose them to the reasons why we want and need mathematical knowledge.  

But while they're building towers with Legos and creating origami snowflakes, don't forget to teach them to make change, calculate a tip, and choose a cell phone plan.  We need those skills too, maybe even more.