Counting Principles show the progression of counting and quantity. The counting principles divide counting and quantity into five basic steps. This is essential for developing number sense. Children may seem to learn to count in the same way they learn the alphabet- rote repetition. Teaching rote repetition of numbers in their correct order is different from developing a correct number sense. In order for students to develop the flexibility necessary for higher order math, students need to develop a strong understanding of the relationship between counting and quantity.
Counting Principles
1. Stable Order
Stable order means that a student is able to recite the numbers in a correct order. Strategies that Support Student Learning
Putting pictures of items in order from smallest to largest, based on quantity, and counting them forwards and backwards.
Organizing objects in order, without numbers at first, then adding the symbols later.
If you observe: A child miscounting orally by rote or with objects… Consider: Intentionally miscounting and ask the child to tell you what number you missed.
2. One-to-One Principle
Understanding that each object in a group is counted once and only once. In early stages, students will touch or tag each object and say a label. In order for students to understand one-to-one correspondence, children must be able to rote count. This can be promoted by counting objects during play and everyday life. Children may also show quantities on their fingers. Strategies that Support Student Learning
Encouraging students to “tag” or move items out of the way while counting.
Matching items with pictures. For example, using search and find books.
Encouraging students to create a tally chart to count and track the quantity of food, toys, sounds (i.e.: taps on a drum), letters in a word or words in a sentence.
If you observe:
A child playing in the kitchenette area preparing food for stuffed animals…
Consider:
Asking how many items of food they are preparing or how many people they are cooking for.
3. Cardinality Understanding that the last number counted a group of objects represents how many objects are in the group. If you are struggling to assess whether a student firmly grasps the cardinal principle, consider asking the student to count a group of items and then ask them to put the same quantity into a bag. If they must recount, they may not have a firm understanding of cardinality. Strategies that Support Student Learning
Encouraging students to show you a group of items to match a specific number.
Ask students to count a group of items in a set. Then, explicitly ask them to show you how many objects in that group represent that amount.
If you observe:
A child building a tower out of blocks
Consider:
Asking if they can use the same number of blocks to create a path.
4. Conservation Understanding that the count for a set group of objects stays the same no matter whether they are spread out or close together. If a student counts a group of items that are close together and then needs to recount after you spread them out, they may not have developed an understanding of the principle of conservation. It is all too common that we rush towards symbols in mathematics and counting is no different. Help children develop a firm grasp of the quantity associated with each number concretely before we formally introduce the symbolic form of number.
5. Order Irrelevance The order in which items are counted is irrelevant. The child is able to start at different points in the group when counting. For example, counting from the left-most item to the right-most and visa versa. Many students do not fully develop order irrelevance until 4th grade. Explicitly teaching this principle is important. It should also be noted that just because a child is strong in this principle, they may still be weak in other counting principles. Strategies that Support Student Learning
Counting sets of items from left-to-right, right-to-left, top-to-bottom and bottom-to-top.
Counting sets of unique items (different color, shape, etc.) in a variety of orders.
If you observe:
A child counting a set of toy cars…
Consider:
Asking if they can predict how many cars there would be if they started counting from a different spot.
6. Abstraction Abstraction requires an understanding that we can count any collection of objects, whether they are seen or not seen. For example, the quantity of five large items is the same count as a quantity of five small items or a mixed group of five small and large things. Children often consider groups of larger items to have more value than groups of smaller items. For example, a child may believe that the quantity of the 3 cars in the parking lot is larger than the 3 toy cars placed on the play mat. Strategies that Support Student Learning:
Counting non-tangible quantities such as sounds, actions, words, questions or steps.
Matching groups of different items with the same quantity.
If you observe:
A child playing with toys of different sizes…
Consider:
Taking a group of 2 larger items and a group of 3 smaller items and asking which has more.
Sources:
Bermejo, V., Morales, S. and Garcia de Osuna, J. (2004) Supporting children’s development of cardinality understanding, Learning and Instruction, 14: 381–98. [PDF]
DCSF (Department for Children, Schools and Families) (2009), Children thinking mathematically: PSRN essential knowledge for Early Years practitioners. Nottingham: DCSF. [PDF]
Gelman, R. & Gallistel, C. (1978) The Child’s Understanding of Number. Cambridge, MA. Harvard University Press. [PDF]
Clements, D. & Sarama, J. (2010) Learning Trajectories in Early Mathematics – Sequences of Acquisition and Teaching. Buffalo, NY. [PDF]
12, K. P. on J., 21, K. P. on D., 14, K. P. on M., 20, K. P. on J., & *, N. (2021, February 27). Counting Principles - Counting, Quantity and Cardinality. Tap Into Teen Minds. https://tapintoteenminds.com/counting-principles/.