One night before bed, in the spirit of Piaget, I decided to videotape my 2-year old daughter counting. As most parents know, counting is a fundamental part of children’s everyday mathematics experiences.

Here’s an example of my daughter counting. She’s starting to show understanding of certain aspects of counting and patterns. Pay close attention to the end of the video:

Before kids slide down a slide or begin a race, they chant, “1,2,3, go!” Kids learning to count often spend what feels like hours counting everyday objects, like fingers and toes, cheerios for breakfast or stairs they are climbing.  Children learn counting as if it were a song, a rhythmic chant.  Eventually, they begin to see patterns in the song that they can extend.

Young kids are often able to count higher than you might expect if you help them with a few key numbers, particularly the decades.  For example, a 2.5-year-old might count up until 19 and then get stuck on what comes next, but after you tell her it is 20, she can keep counting until 29.  If you tell her 30 comes next, she will keep going until 39, etc. In a past blog post, I wrote about how language makes our counting words particularly tricky for English-speaking kids, especially compared with their Asian peers.

But knowing “the counting song” is just the tip of the iceberg in learning how to enumerate, or count objects. Researcher Rochel Gelman from Rutgers University outlines 5 key principles that underlie the ability to enumerate.

1. Stable ordering principle: This is what most of us think of when we say, “My child knows how to count.” It refers to knowing the counting sequence (or the “counting song”). But knowing the count sequence doesn’t necessarily mean you can count objects or use the sequence in any meaningful way.

2. One-to-one principle: Each object to be counted gets one and only one counting word. In other words, no double counting jelly beans and no skipping objects when you count.

3. Cardinal principle: Knowing that the last number you say when counting objects refers to the entire set of objects, not just the last object. This is one of the hardest principles for little kids to grasp and has been the subject of much research. The classic Piagetian task testing this principle is to ask a 3-year-old to count a set of blocks. The child may be very good at carefully counting, “1,2,3,4” as he points to each of the 4 objects; however, if you then immediately cover the objects with your hands and ask how many objects he just counted, the child doesn’t know how to answer. He needs to count them all over again, starting from 1 because he doesn’t realize that the “4” he just said refers to the set. One way to help reinforce this idea with kids is to have them always state the cardinal value after they count, i.e. “1,2,3,4. 4 blocks.”

4. Abstraction principle: You can count just about anything (blocks, candies, ideas or all three mixed together)

5. Order-irrelevance principle: Items can be counted in any order. This idea can be tricky for young kids, too. If you line up blocks in a row, many kids will think you have to start from one specific end and not from the other end or even from the middle of the line.

Here’s an example of my daughter demonstrating the order-irrelevance principle:

Importantly, it is completely normal for kids to make these kinds of counting mistakes. The important thing is to keep encouraging your kids to count and to keep it fun. If you can come up with a fun way to challenge their misconceptions, GREAT! But if not, it’s OK for kids to keep making these mistakes. With practice and experience, they will learn these principles.