# Why We Suck at Division & Subtraction Inherently

Disclaimer: I am not at all a neuroscientist. I’m just a guy that has an interest in it and saw a pretty cute connection.

Of the different forms of arithmetic, I believe that subtraction and division, which if we were being honest is just a glorified form of subtraction, are the most difficult manipulations to perform $-$ why, you might ask, well I believe the reason is due to object permanence.

In a study published in the Journal of Experimental Psychology, titled Cognitive Arithmetic Across Cultures1, compared the performance of both basic and advanced arithmetic across certain cultures. The study inspected how people from Asian Chinese, Canadian Chinese, and Non-Asian Canadian backgrounds performed in Zero and One problems. Zero problems, for example, are those which the operations are as follows

\begin{align*}\text{Addition: }& x+0=x, 0+x=x\\ \text{Subtraction: }& x-0=x, x-x=0 \\ \text{Multiplication: }& 0\times x=0, x\times 0 =0\\ \text{Division: }& 0\div x = 0. \end{align*}

Meanwhile, One problems represent the following procedures

\begin{align*}\text{Addition: }& x+1=y, 1+x=y\\ \text{Subtraction: }& x-1=y, x-y=z \\ \text{Multiplication: }& 1\times x=x, x\times 1 =x\\ \text{Division: }& 1\div x = x, x\div x =1. \end{align*}

Before we continue notice that only multiplication and division in One problems return either $1$ or $x$, while addition and subtraction involve a different number. This distinction causes addition and subtraction to be solved by recalling previous information, or performing a procedure, such as counting, and therefore makes these more difficult to complete.

Although the aim of the study was to determine arithmetic performance across cultures, looking at the data there is a clear indicator that subtraction and division were the most time consuming arithmetic operations to perform. In fact, Table 8 of the study illustrates that perfectly; Who knows, perhaps this trend does not hold over a larger sample size or with the inclusion of more ethnicities. But, if we were to take the data for its word, the question still remains, why? Why do division and subtraction take the longest to perform?

##### Object Permanence and Infants

Do you remember playing peek-a-boo with your parents?2 Well, neither do I. But the act of playing peek-a-boo with an infant reveals that, when you hide your face, the infant seems to believe that you have actually disappeared. Basically, the infant hasn’t learned about object permanence, or the understanding that objects continue to exist even when they cannot be observed. While the process that an infant undergoes in order to learn object permanence is amazing, what is really interesting is the connection between object permanence and mathematics.

In fact, this idea is not unique. A simple google search reveals that people, in fact, have connected object permanence and mathematical development before.  However, I would like to assert that the difficulty of performing subtraction and division can be attributed to the presence of object permanence in infants.

For instance, imagine an infant in front of $x$ objects. Then if you were to remove one object, so that you have $x-1$ items left, then not only does the infant need to learn that the removed object still exists but the infant will eventually need to index the $x-1$ objects with a variable $y$, which is a difficult task to do as we’ve seen in the study. My hypothesis is that subtraction and division, especially the operations found in the One Problems, is akin to removing an object from the infant’s field of vision, and the performance on subtraction and division serve as a residual of the time it takes to grow into object permanence.

#### Footnotes

1. Journal of Experimental Psychology: General 2001. Vol. 130, No. 2, 299-315
2. If you said yes, you’re a dirty liar because permanent memories don’t form until around 3 years old and by that point your parents wouldn’t be playing peek-a-boo with you.