Station  J

Piles and Holes 

Here’s what led me to this belief. It comes from another untrue story.

As a young child I used to regularly play in a sandbox. And there I discovered the positive counting numbers as piles of sand: one pile, two piles, and so on. And I also discovered the addition of positive numbers simply by lining up piles. For example, I saw that two plus three equals five simply by lining up piles like this.

I had hours of fun counting and lining up piles to explore addition.

But then one day I had an astounding flash on insight! Instead of making piles of sand, I realized I could also make holes. And I saw right away that a hole is the opposite of a pile: place a pile and a hole together and they cancel each other out. Whoa!

Later in school I was taught to call a hole “$-1$”, and two holes “$-2$,” and so on and was told to do this thing called “subtraction.” But I never really believed in subtraction. Although my colleagues would read $5 - 2$, say, as ”five take away two,” I was thinking of five piles and the addition of two holes. A picture shows that the answer is three piles.

Yes. This gives the same answer as my peers, of course: all correct thinking is correct! But I knew I had an advantage. For example, my colleagues would say that $7-10$ has no answer. But I could see it did.

$\begin{aligned}$
7 - 10 &= \text{seven piles and ten holes}\
&= \text{three holes}\
&= -3\
\end{aligned}

(By the way, I will happily write $7-10$ as "$7+-10$". This makes the thinking more obvious.)