Questions: Counting a Set of Objects Up to 20

Question 1

A child counts 9 toy bears and arrives at the number 9. Then the teacher rearranges the bears into two piles. The child is asked, 'How many bears are there now?' What does a child with the cardinality principle do?

• A. Count all the bears again from the beginning
• B. Say '9' immediately without recounting
• C. Count just one pile and double it
• D. Say they don't know because the bears moved

Answer: (B)

The cardinality principle means the child understands that the last number said when counting names the total. Moving objects doesn't add or remove any, so the total is still 9. A child without the cardinality principle treats counting as a performance that must be repeated whenever anything changes — they haven't yet grasped that the final count-word is a property of the whole group.

Question 2

A child counts 12 blocks by starting at the left side of a row, getting 12. Her teacher asks her to count the same blocks starting from the right side. What should happen, and why?

• A. She gets a different number — counting direction changes the total
• B. She gets 12 again — the count does not depend on where you start
• C. She should not recount — once is enough to be certain
• D. She gets 12, but only if she moves the blocks first

Answer: (B)

This is the order irrelevance principle: as long as every object is counted exactly once, the starting point and direction don't affect the total. Experiencing this by actually recounting in different orders — and always getting the same number — builds deep number sense. Option A represents a common early misconception; option C reflects the cardinality principle but misses the educational value of discovering order irrelevance.

Question 3

If you count a group of objects in two different orders and count each object exactly once each time, you will always get the same total.

• T. True
• F. False

Answer: True

This is the order irrelevance principle. What matters is that every object is counted exactly once — not the sequence in which they are counted. Discovering this through hands-on experience is one of the key ideas in early number sense development.

Question 4

After a child counts 14 blocks correctly, moving those blocks into a circle means the child is expected to recount them to know how many there are.

• T. True
• F. False

Answer: False

Moving objects does not add or remove any, so the total stays the same. Recounting is only necessary if objects are added or taken away. The cardinality principle tells us that once we have a valid count, the last number said names the set — and that doesn't change when objects are rearranged. Children who feel compelled to recount after every rearrangement haven't yet internalized this principle.

Question 5

What is the cardinality principle, and how can a teacher quickly test whether a child has internalized it?

Model answer: The cardinality principle states that the last number word said when counting a set tells you the total number of objects in that set. A teacher can test it by having the child count a group of objects, then immediately asking 'How many are there?' — without moving anything. A child who has the cardinality principle answers immediately with the number they just said. A child still developing it will recount, suggesting they treat counting as a procedure rather than a way of finding a quantity.

This distinction matters because cardinality is the bridge between the counting sequence (reciting number words) and number meaning (understanding what those words represent). Counting is not just a performance — it produces information about the size of a set.