We are often unaware of how intricate the mechanisms that govern our lives can be. Many of the things we do without thinking seem deceptively simple, but actually, they're anything but that. For example, scratching your nose actually requires a complex set of lightning-fast interactions between your brain, as well as the muscles in your arm and hand. Our lives revolve around numbers, but counting is probably something most of us never think twice about. However, the way our minds process numbers is unexpectedly interesting. To illustrate, here's a question:
How many apples are there here?
How about now?
Did you notice a difference in how you got to your answer? For the first group of apples, you were probably able to tell how many there were right away. The second group might have taken you a little longer. Maybe you counted the apples individually, or divided them into smaller groups and added those up.
Why did this happen? The difference lies in how we process numbers. The human mind has two systems for counting: one for numbers up to four, and another for numbers greater than four. The first one is called the subitizing system. It's a cognitive system that gives us the ability to make rapid and accurate assessments of a number of items directly in front of us, with one catch: It only works if you have four or less items. [1] The name comes from the Latin word subitus, meaning “sudden”.
The second system is called the approximate number system. It's a cognitive system that allows us to estimate the number of a large group of items, usually with much less precision. If you say that there are about 50 people in the cafeteria, you are estimating. Let's say that you have a group of 8 students and a group of 10 students. Without
counting, you would be able to determine which group is larger with the same accuracy as you would distinguish between a group of 80 and 100 students. [2] This is because the ratio of both groups is 4:5. Unlike subitizing, estimation abilities do differ slightly from person to person and may be linked to better math performance. [3]
So, if our subitizing systems stop at four, how are we able to accurately count larger groups of items? As you can see, numbers are not inherently natural to us- no one is born with a predisposition for counting (So don't feel bad for being bad at math!) What does enable us to count accurately is language. At first, when we learn to count as kids, numbers are just a long string of words to be memorized, much like the alphabet. With time, we begin to understand that numbers represent quantities, and each successive number in the sequence represents a quantity that is larger than that of the previous one.
Back to our example: when you were shown the first four apples, you were immediately able to tell how many there were by using your subitizing system. The second group of apples, however, was too large to be counted this way. If you want to see how many apples there are, you have two options. The first would be to enumerate: you give every apple a number from your list and stop when you run out of apples. In this case, you would start with one apple and count: “One, two, three, four...” until you arrive at six, and now you know that there are six apples. The second way would be to split the group into smaller, subitizable groups of four or less, like two groups of three, and then add those up. Splitting groups of numbers this way still falls under the category of subitizing, however, if you want to count a very large amount of items, you would have no choice but to use your counting system.
Interestingly enough, not all cultures have words for numbers the way we do. For example, the Munduruku, an indigenous Amazonian people, only have words for numbers up to five. [4] In addition to that, human beings are not the only ones with this kind of number sense, or numerosity. Studies have shown that some animals and insects, like fish [5], ants [6], and ravens [7], also possess numerical and subitizing abilities.
One application of subitizing is digit grouping. Most of the long strings of numbers or characters that we encounter in everyday life, like phone numbers, addresses, postal codes, and bank card numbers, are typically divided in groups of 3 or 4 and separated by dashes, spaces, or commas. Visually separating long strings of numbers makes it easier for us to recognize and remember them. Compare 1000000 to 1,000,000. The latter, with the zeroes split into smaller, subitizable groups, is instantly more recognizable. Without the commas, the zeroes all blend together. Dice, playing cards and other gaming tools also split quantities into subitizable groups with recognizable patterns.
Isn't it interesting how the numbers around us are arranged so that we can more easily recognize and remember them? Perhaps now that you're more aware of how you count things, you've gained an appreciation for the systems of the human mind, and maybe started thinking more in-depth about the other things you do. If you're curious to see how good your estimation skills are, you can head over to www.panamath.org/testyourseIf.php and take the
test offered.
References:
Jevons, W.S. (1871). The power of numerical discrimination. Nature. 3 (67): 281—282 Sousa, David
(2010). Mind, Brain, and Education: Neuroscience Implications for the Classroom. Solution Tree
Press.
Libertus, Melissa E. (2011). Preschool Acuity of the Approximate Number System Correlates With
School Math Ability. PMID 22010889
Pica, P., Lemer, C., Izard, V. & Deh (2004). Exact and approximat
in Amazonian indigene gr 499-503
Agrillo, Christian (201 Numerical Systems mans and Guppie Reznikova, Zhan petence in anim
Behaviour, 148(4) 405-434