How do children learn numbers?
This semester, I am teaching ݮƵ teacher candidates about elementary mathematics education. This has given me an opportunity to revisit, reflect and extend my own thinking about early mathematical development. One of my recent lectures focused on the development of numerical reasoning and I was reminded how fascinating and shockingly similar all humans are when it comes to the early development of numerical skills.
Numerical reasoning is the ability to make sense of quantity – interpreting, comparing, and operating with numbers. At a very young age, children often learn the rhythm of number words before those words carry meaning. For example, children will chant numbers to the rhythm of walking up the stairs, sing songs with numbers, and tell you “I want five more minutes” without understanding the weight these numbers carry. I recently saw this firsthand when I was at the park with my four-year-old son and told him he had one more minute to play. He demanded 10 minutes and when I responded with a counteroffer two more minutes, he smiled and said, “Yes,” believing I had given him significantly more time to play.
Research shows there is a clear developmental path to numerical understanding. First, children are “one-knowers” meaning that they understand the meaning of number one. Children then move to understanding the number two, then three and ultimately number four. After that, children seem to make a developmental leap in their understanding and can apply their reasoning to all higher numbers (Sarnecka, 2015).
This developmental path begins with a set of core principles where children gradually move from reciting number words to understanding number as quantity.
5 core principles of numerical skill development
Stable order principle
The first principle, seen very early, is a child’s ability to recite numbers in a fixed, repeatable order. Children may not yet understand what the numbers mean, but they recognize that number words belong in a particular sequence. For example, they know the number three comes before four and so on. Many children delight in chanting these words, sometimes counting toys or counting down from 10. Even when their recitation is incomplete or appears to be jumbled, it shows an early grasp that numbers carry structure and meaning. For example, it's amazing how a child can show they understand the structure of counting by reciting, "11, 12, 15, 19, 20." While they may not recite all of the numbers, they demonstrate significant principles of counting. The next time you’re with small children ask yourself if and how they are reciting number sequences during play. Are they skipping or repeating numbers?
One-to-one principle
Once children can say some numbers in order, they begin to connect those number words to the objects they are counting. This demonstrates a one-to-one understanding that each object corresponds to a single number word. At first, children may point to or touch each item as they count, but they often skip items or count the same item more than once, indicating the skill is still developing. Over time, however, they become more deliberate, often inventing or learning strategies like moving objects into a separate pile or touching each finger to their chin to keep track and ensure accuracy.
Order-irrelevance principle
The order-irrelevance principle is the understanding that objects can be counted in any order and the total will remain the same. For example, whether a child begins counting blocks from the left, the right, or the middle, the final count should not change. Early on, children may be uncertain about this and need to practice by starting from the same place each time. With experience, they gain confidence that the order of counting does not affect the result.
Abstraction principle
As counting develops further, children understand the rules of counting apply no matter what is being counted. Known as the abstraction principle, this stage indicates an ability to count not only physical objects but also sounds, actions, or even ideas. They are beginning to see counting as a flexible tool that can be applied to anything.
Cardinal principle
The cardinal principle is the final major milestone in the development of early counting. It is the understanding that the last number spoken tells “how many” are in the set. For example, when a child counts five blocks and then says, “there are five,” they are demonstrating that they recognize the total quantity, not just the sequence of number words. Mastering this principle lays the foundation for more advanced ideas in mathematics, such as comparing sets, adding and subtracting, and eventually working with place value and number operations.
If you work with children, you know the development of skills progress differently for each child from one stage to the next. Children ebb and flow among the stages, in one moment showing proficiency and in the next showing development. The key marker of proficiency is when you notice consistency in how children are numerically reasoning.
Interestingly, children all around the world seem to acquire numbers in remarkably similar ways. Researchers studying numerical development across languages, cultures and countries have found that learning to count emerges in the same sequence for all children (Piantadosi, Jara-Ettinger, Gibson, 2014). While there are differences in the ages that children acquire these skills, the underlying trajectory is universal.
So how can we provide rich numerical environments to support children’s development? Draw children’s attention to all the numbers embedded in their environment, adapt songs to count in different ways (e.g., forwards, backwards, leave numbers out in a sequence), and PLAY! A positive association and interest in number will support children’s natural curiosity to learn more.
Give-N Task
Robertson activities supporting numerical reasoning
Ladybug Houses
Number Hunt
Making Clocks
Unifix Counting Towers
Story Problems
Target Number Game
References
Piantadosi, S. T., Jara‐Ettinger, J., & Gibson, E. (2014). Children's learning of number words in an indigenous farming‐foraging group. Developmental Science, 17(4), 553–563.
Sarnecka, Barbara. (2015). Learning to represent exact numbers. Synthese. 198.
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