I have heard the claim “calculus is easy, algebra is difficult, and arithmetic is impossible,” but if that is true, then what does that make counting?
We often hear little ones proudly singing the alphabet song or reciting a string of numbers from 1 to 20. Have you ever asked one of those who now “knows their ABCs” what comes after N? Or D? How about what comes after R? They don’t know, do they? Or maybe they can answer your question, but only after starting the song over again from A.
Imagine that that you have a friend who has 12 children. He tells you the names and ages of all of his children. How easily would you be able to name his children and their ages? Most likely you would start the learning process by reciting the children’s names and ages in a specified sequence. But then imagine that he begins talking about his eight-year old and you suddenly struggle to remember that child’s name.
The examples listed above show that even a simple sequence can be challenging for us when, or if, we are asked to isolate a portion of it. More often than not we run through the entire sequence to find our way. This is similar to what children experience when they begin learning to count. A beginning counter cannot answer questions about what comes before or after a particular number, much less figure out three more or three less. At this point in their development it is much like learning the alphabet. Eventually, they attach a word (“six”) with a symbol or image (“6” or the dots on dice), but these number words and images don’t have numerical value for them.
Even before counting happens, there are some very special “numerical” experiences. I put “numerical” in quotes because the experiences may appear numerical to you and me, but do not possess numerosity from the point of view of the child. Two is a wonderful number. Children learn the idea of “one” early on. And “two” follows because the have many experiences with duals or pairs. They have two feet, two eyes, two ears, and two hands. And with those two hands they can pick up two toys, two carrots, two shoes, etc. At this point, everything beyond two is simply “many.” Too many. They no longer have enough hands to pick up each item.
“One” and “two” become the child’s first numerical experiences. Sometime after that patterns become an important element in the child’s development of counting. For example, a five pattern might be identified by a hand or a dot pattern (see figure 1). They begin to recognize patterns for 2, 3, 4 and 5 (see figure 2). But make no mistake, these are images, not numbers for the child. They simply associate a single word with an image, much as they would for “cup” or “dog.” The child is unaware of the individual components that make up the image. They will recognize them before they can count them. In some cases, like the Mayan number system (see figure 3), a non-composite image for five can help extend the recognizable patterns such as six is five and one, seven is five and two, etc.
In my next blog, I will continue exploring the nature of counting and explain why I have a newfound respect for children learning to count and the level of mathematical sophistication that counting requires.