Wednesday, March 5, 2014

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.  


  1. Deb, I love your concise summary, and your thoughtful questions. These themes need more discussion:

    - Once kids experience free play ("This is cool" stage), where do they go to work on patterns of calculus, and to formalize their findings?
    - If learning mathematics becomes more nonlinear (with children working on mathematical subjects early on), how do they develop the foundation skills of number, measure, and space?
    - What about science, technology, and everyday life applications?

    Maybe we can talk about these three issues in a live conversation, such as a webinar or a Google Hangout? I would like to learn from you.

    1. Wow... I never imagined the real Maria Droujkova would comment on this post. I am honored... thank you! I posted The Atlantic article on a site for calculus teachers, but did not get a lot of comments. Your ideas challenge what math teachers do every day and although the ideas are interesting and could perhaps change the face of mathematics education in America, they also require answers to the questions you ask me here. I think your questions are really important, ones I don't think I can answer alone. I agree that some students have negative experiences through the drill and kill of arithmetic, but at the same time, we as high school teachers find ourselves repeatedly frustrated by the fact that our students can't really fly with abstract concepts because they are still stuck trying to find the factors of 64. At some point they must simply know multiplication, and I don't mean "figure it out." We spend a lot of time in elementary school trying to teach them multiplication with blocks and all manner of concrete and kinesthetic tasks, but at some point, 7x7must be a given, a number that is as obvious as spelling the word "the." Otherwise we are never going to get past to the basics so we can really explore all that stuff they thought was cool in 1st grade. So I return to your questions with the sure knowledge that I am not sure I can teach you anything. I find that the best laid plans are those that have a the goal in mind and so I guess all conversations should begin there. What is it that we want people to know about math by the time they finish school? Once we have the end result determined, we can start to talk about the various roads we might use to get there.

      Thank you so much for stopping by and for your thoughtful words. I am barely involved in google these days but will check out how I might establish a hangout. Pehaps you already know how and we can plan a time to do so and invite some other folks to chime in on the conversation. I look forward to continuing this conversation.

    2. If you two do have a conversation on a webinar, can I be invited also. I am also a math teacher and a product of drill and kill. But I also love frank and honest discussions about topics such as this.

    3. Absolutely... I never heard back from her...

    4. Sorry about the delay, Deb! I have not forgotten, I am just trying to figure out how to go about this. Do you have other teachers you would want to invite to the discussion? What can be our goal for the meeting? How about forming a list of questions that help people clarify their choices of what they see as the end result of secondary math ed?

      This can be similar to questionnaires that help people map answers to small questions onto bigger categories. I've seen several that can "measure" your affiliation. So we can think of questions that people can ask themselves, to measure their stance about math ed. Does it make sense?

    5. I think I need more details as to what you are suggesting. Can you show me an example of the kind of questionnaire you want to use? Would you administer it in advance or during the discussion? Are you suggesting the answers might guide the discussion or would you have a preset list of questions and we would work our way through them over the discussion. And I am still not sure how the meeting/hangout would work. Do we still need to figure this out? Lots of questions... :)

  2. Let us make this conversation happen, with a simple mode that does not require preparation. We will do it within the Math Future series, which we are re-launching this August: The webinar works by downloading a Java program that runs in the browser (BlackBoard Collaborate). You would invite the colleague(s) you had in mind, and we will discuss this as the central issue: "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." I want to ask questions about your perspective, as a high school teacher, on what parents and teachers can do when their children are five. My email is for details. I am very much looking forward to talking with you!

    1. I am sorry about the delay in responding. I shut down this blog for the summer and have been busy getting ready for the start of school next week. Please give me more information about the types of people you think should attend this "chat" and what exactly it would entail. Do you have a date in mind?


    2. Deb,

      We usually send announcements about upcoming events out to parents, teachers, and math ed developers who follow Natural Math and Math Future projects. So, the audience is general. The idea is for presenters to talk about issues of mathematics education, and for the audience to ask questions and offer comments. The conversations are friendly and open: it is about celebrating ideas we love. We will start events at the end of August, so September or October are open, during weekdays (Monday-Thursday). We can pick the time convenient for you.