Tuesday, May 22, 2012


Dog bloggers find dog bloggers.  And I found Paul Handover who blogs at Learning from Dogs  Paul fell in love with dogs when he was ten, but circumstances didn't allow him to have a pooch of his own for another 40 years, after he'd retired from a successful career in sales and company management.

This is Paul's dog Pharaoh, "Just being a dog."
Now Paul has 11 dogs.  (Let this be a lesson to all parents who prevent their children from having a dog - One pooch will never be enough and eleven is just a start.)  Paul's blog isn't so much about dogs as it is about learning from dogs who obviously have things figured out, and one thing they've figured out is math.

This photo proves Beagles can do basic math and a little algebra.

But Corgis can do calculus. 
This is a Tim Pennings' Corgi with his calculus tools.

Paul told me about Tim Pennings, a professor of mathematics at Hope College who specializes in research and writing about dynamical systems, mathematical modeling, and the infinite.  Dr. Pennings lives with a famous Pembroke Welsh Corgi dog, Elvis, who demonstrates calculus.  Dr. Pennings and Elvis are featured here - A Dog, a Ball, and Calculus, Ivars Peterson's Math Trek (copyright 2003).

"Depending on the angle at which Pennings tosses the ball relative to the shoreline, his dog can run along the beach until he is directly opposite the ball, then swim out to get it. Or Elvis can plunge into the water immediately, swimming all the way to the ball. What happens most of the time, however, is that Elvis runs part of the way, then swims out to the ball. This behavior reminded Pennings of a standard problem found in just about any calculus textbook—one that involves minimizing the time of travel to a target when the available paths require traversing different mediums at different speeds." Professor Pennings said, "Of course, although he makes good choices, Elvis doesn't actually do calculus. Nonetheless, Pennings remarked, "Elvis' behavior is an example of the uncanny way in which nature . . . often finds optimal solutions. "I'd guess that most dogs have the same problem-solving software built in from the factory," he says. My article is just "drawing attention to something that has been in front of us all the while."

Walk softly and carry a big stick.  
I'm sure there's some calculus going on here, but I don't know what.

Dr. Pennings has given hundreds of talks with Elvis. His areas of research and writing include dynamical systems, mathematical modeling, and the infinite.  Click here to learn more about calculus, how dogs use it and the man who lectures about it:  Hope College Mathematics

And even more links:
Do dogs know Bifurcations?  (Tim Pennnings and Roland Minton,  The College Mathematics Journal, Mathematical Association of America)

Tim Penning with his teaching assistants.

And don't forget to check out Paul's blog, Learning From Dogs


  1. That was thought invoking, it is amazing how animals, just seem to know.
    My dog's greatest math skill is, how many sticks he can fit in his mouth and still leave room for a tennis ball.


  2. Deb- I think that is algebra II.

  3. Woof Woof from Elvis' sister Dee Dee and Aunt Princess.