Saying Numbers Is Not Understanding Them: What One-to-One Correspondence Means

By about age three, many children can say “one, two, three, four, five … ten” in the bath. But put five oranges on the table and ask “How many?”, and a child may point along and count up to six. Or, after counting, when you ask again “So how many?”, they start counting again from the beginning.

The child is not being silly. Between saying the numbers and understanding them there are several steps. In this post I write about those steps, based on what developmental research has found, and about games you can play at home. It is about children who have not started school yet.

Five principles of counting

Rochel Gelman and Randy Gallistel, who studied how children count, set out five principles needed to count objects correctly in their 1978 book The Child’s Understanding of Number. Their list is still widely cited in research on number development, and it is discussed in a 2008 paper by Barbara Sarnecka and Susan Carey.

1. One-to-one. Match exactly one number word to each object, without missing any or counting any twice
2. Stable order. Always say the number words in the same order
3. Cardinality. The last number said tells how many there are altogether
4. Abstraction. You can count anything: oranges, sounds, or a mix of different things
5. Order irrelevance. Whichever object you start with, the total stays the same

Reciting “one, two, three … ten” shows the second principle, stable order. The other four may not be there yet.

One-to-one correspondence

Counting five oranges as six often happens while the one-to-one principle is still shaky. The finger runs ahead and the voice falls behind, or the voice runs ahead and the same orange gets counted twice, or one at the end is skipped.

One-to-one correspondence can actually be practised without number words at all. Lay out plates and put one spoon on each. Sit one person on each chair. Put each pair of shoes in its own cubby. Each of these links two groups one by one. It is good for children to have plenty of this “one each” feeling in daily life before counting.

When counting, moving each counted object to one side works better than pointing. It cuts down on skipping and double counting, because the counted and uncounted objects can be seen apart.

The last number is “how many”

A child who counts again when asked “How many?” may not yet have the third principle, cardinality. They know counting as a ritual of saying number words in order while pointing.

In a 1990 study, psychologist Karen Wynn asked children of about two to three years old to recite numbers and to hand over a requested number of objects, as in “Can you give me three?” Many could recite and point well before they could hand over exactly the number asked for.

Later research suggests that children learn the meanings of small number words one at a time, first “one”, then “two”, then “three”, over many months, and only after that work out that “you count, and the last number is the answer”. Sarnecka and Carey’s paper looked at this sequence. In studies of American children, this usually happens somewhere around three and a half to four, but children vary, and there is no age by which it must happen.

Same number, different arrangement

There is another well-known step. Lay out two rows of counters with the same number in each, and once the child agrees they are the same, spread one row out. Young children often say the longer row has more. Jean Piaget called this conservation of number, and he thought children understand that the number does not change with the arrangement at around six or seven.

But the result depends partly on how the question is asked. In a 1974 study by James McGarrigle and Margaret Donaldson, when a “naughty teddy” messed up the row by accident instead of the adult moving it on purpose, many more children answered correctly. Children may have assumed that if an adult deliberately rearranged the row, something must have changed.

The fifth principle, order irrelevance, is part of the same understanding: the number does not change with how you arrange or count the objects.

Numbers you know without counting

Up to about three, people can tell how many there are at a glance without counting. This is called subitizing. When objects are arranged in familiar patterns, like the dots on dice or the marks on playing cards, five or six can also be seen at a glance.

Counting and seeing at a glance grow as separate abilities. Small numbers are seen at a glance, larger ones are counted to check. Being able to move between the two leads to ways of seeing numbers such as “five is three and two”.

Using fingers

People sometimes ask whether children should be made to stop counting on their fingers. Before school, there is a view that children can use them as much as they like.

Fingers are a one-to-one tool you always carry. Fold down one finger at a time while counting, and the counted and uncounted are kept apart. Five on one hand and ten on two also give a pattern to see at a glance. There is plenty of time later to manage without fingers. I also wrote about preparing for school math in Getting Ready for School Math.

Number games at home

Here are the steps above turned into games at home.

Handing out. Give each family member one snack or one sheet of origami paper. This practises one-to-one correspondence. When you finish, ask, “How many did you hand out?”

“Give me” games. Ask, “Can you bring me three oranges?” This is Wynn’s “give me three” done at home. When they bring them, count them together to check.

Rearrange and ask again. After counting five blocks, change the arrangement and ask, “How many now?” If the child says “five” without counting again, order irrelevance and conservation are taking hold.

A board game with numbered squares. In a board game where the squares are numbered in order from 1, the child says a number with every move, and the order and size of numbers are felt through play. In a 2008 study by Robert Siegler and Geetha Ramani, preschoolers who played a few sessions of a game with squares numbered 1 to 10 in a straight line got better at estimating where numbers go on a number line.

Stairs and bath time. Count each step on the way up, or count to ten before getting out of the bath. Counting in the same situations every day practises stable order.

A few minutes at a time is enough for each of these. How long children aged three to six can focus is in How Long Can Preschoolers Focus?, and how to connect this to writing numbers is in How to Teach Kids to Write Numbers.

Counting games in BrainDojo Kids

BrainDojo Kids is an iPhone and iPad app for ages three to six that I make. It has several training games about counting.

In the Count It game, various pictures are scattered across the screen. The child is asked “How many in total?”, or is shown one kind of picture and asked how many of that kind there are. They count and write the answer by hand. Questions where you count only one kind among mixed pictures are practice in deciding for yourself what to count. This game appears when the age is set to four or older.

In Count Fingers the child looks at a hand and counts the fingers, and in Count Balls they count balls, then write the number. With the age set to three there are easier versions up to five, Easy Count Fingers and Easy Count Balls, and from four there are problems up to ten.

In Sugoroku Together, a board game for the whole family, landing on certain squares starts a mini game. One of them, Count Together, has everyone point and count a crowd of things together and then write the answer by hand. There is also Finger Count Battle, where two players sit facing each other, count fingers, and whoever writes the right number first wins.

The three daily games in Today’s Training, the four Recommended games and Sugoroku Together are free, and every other game can be tried once. A Premium subscription (monthly or yearly) opens all courses and games. In the parent settings there is a daily time limit, on by default at 10 minutes. There are no ads and no account to create.

Getting Started with BrainDojo Kids covers the first session. After counting in the app, try counting real oranges or counters the same way. That helps the numbers on the screen connect with numbers you can touch.