Counting: “Easy as 1-2-3”?

What would you think if I told you counting was actually not as easy as 1-2-3? I was as surprised as you when I studied the development of numeracy in our youngest mathematicians. What we might consider done and dusted counting might just be a memorized verbal sequence of numbers with little meaning. (Sort of like how I used to think I really knew all the lyrics to Queen’s Bohemian Rhapsody.) In reality, counting is far more complex and nuanced–and it is the crucial bedrock for the tower of numeracy that is built through math education… at every stage. It all rests on early counting skills.

Let me start by breaking the idea of counting into its component parts.

Subitizing

Children are born with an innate capacity for quantity. What does that mean? A child can look at a small group of items and know how many there are instantly, without counting. This is called subitizing. As children develop, this ability grows until they can automatically recognize and name quantities up to about 4 or 5 (when a child is around five years old). It works without language, without symbols, without arithmetic. It is simply a sense of how many. Try it for yourself:

From Word Sequence to Counting with Meaning

The ability to subitize gives children their first understanding that counting is more than memorizing and repeating a string of number words. Think of how children learn their ABCs. First comes the song, and with it that endearing "elemenopee." Only later does each letter emerge as its own thing, "L-M-N-O-P...", with its own identity and eventually its own sound. The number sequence works the same way. Children can recite "one, two, three, four, five" long before those words mean anything to them. Being able to recognize a small group of objects at a glance (subitizing) and name it "three" is often one of the first times a number word becomes attached to something real.

One-To-One Correspondence

So, once a child who subitizes understands that these small quantities match a number word, they can begin to count with meaning. The words that make up the forward counting sequence begin to match up, one to one, with objects being counted: one number, one object. When a child counts with one-to-one correspondence, they are tagging each item: matching a number word to a specific object once and only once, with no double counting. The number words also have to come out in the right order every time: one, two, three, four and so on. That is called stable order, and it's what makes the counting sequence meaningful. Additionally, the tag and the word have to land at exactly the same moment! This coordination is called synchrony. One-to-one correspondence is all three of these working together. There is a lot to coordinate in this seemingly simple act. Check out the animation below to see what it looks like!

Cardinality

Okay! We've subitized! We've got the counting sequence down in stable order! We've established one-to-one correspondence! Now what? Now we give real quantitative meaning to the set of objects we've counted. A child has to trust that the final number they say when they've counted a set of objects defines the number of objects counted. If I count five things with one-to-one correspondence, I have five objects, I promise. This is cardinality.

Conservation of Number

Seriously, I promise. Five objects. But here's where it gets interesting. What if your child watches someone move those five objects around, spread them out, bunch them up, mess up a neatly counted pile. And then, that someone asks, "How many now?" If the child can answer with confidence, "five objects, still" without recounting, that child has established the concept of conservation of number. Conservation is crucial: No conservation, no addition. You can't combine two things if you don't trust that either of them will sit still.

Hierarchical Inclusion

This mouthful is actually the piece children bring with them as they transition from the world of counting into the world of additive thinking. What it means is that not only can a child count accurately and with meaning, understanding that the last number counted names the quantity, and that the quantity holds no matter the arrangement, but that child also understands that within those counted items, all previous numbers or quantities still exist. Within five, there is also four, three, two, and one. Numbers are made of all the numbers that came before them.

This seems abstract, but it is what carries students from inefficient counting strategies into additive strategies, where they start to develop real fluency: it all grows from the understanding that numbers are made of...other numbers.

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The good news about all of these technical terms that you don’t need to commit to memory: they are all very easily supported at home in ways that are super fun for all involved. Stay tuned for some lively games and activities to support your expert counters as they work their way toward additive thinking.

References

Clements, D. H., & Sarama, J.(2017/2019). Learning and teaching with learning trajectories [LT]².
Retrieved from Marsico Institute, Morgridge College of Education, University of Denver.

Harris, Pamela Weber. Developing Mathematical Reasoning. Corwin Press, 17 Mar. 2025.

Karp, Karen S, and Barbara J Dougherty. Putting Essential Understanding of Number and Numeration into Practice in Prekindergarten-Grade 2. Reston, Va, National Council Of Teachers Of Mathematics, 2019.

Wright, Robert J, et al. Teaching Number in the Classroom with 4-8 Year Olds. SAGE, 17 Oct. 2014.