Principles Behind Learning to Count
Although I've given names to the concepts behind counting, we don't actually use these names when teaching young learners to count. Rather, we make observations and focus on the concept.
Sequence: Children need to understand that regardless of which number they use for a starting point, the counting system has a sequence.
Quantity or Conservation: The number also represents the group of objects regardless of size or distribution. 9 blocks spread all over the table is the same as 9 blocks stacked on top of each other. Regardless of the placement of the objects or how they're counted (order irrelevance), there are still 9 objects. When developing this concept with young learners, it's important to begin with pointing to or touching each object as the number is being said. The child needs to understand that the last number represents that many objects and that the last number is also the symbol used to represent the number of objects. They also need to practice to count the objects from bottom to top or left to right to discover that order is irrelevant. Regardless of how the items are counted, the number in the set will remain constant.
Counting can be Abstract: This may raise an eyebrow but have you ever asked a child to count the number of times you've thought about getting a task done? Some things that can be counted aren't tangible. It's like counting dreams, thoughts or ideas, they can be counted but it's a mental process, not a physical process where you can touch and count.
Cardinality: When a child is counting a collection, the last item in the collection is the amount of the collection. For instance, if a child counts 1,2,3,4,5,6, 7 marbles, knowing that the last number represents the number of marbles in the collection is cardinality. When a child has to recount the marbles when prompted about how many marbles there are, the child doesn't yet have cardinality. To support this concept, children need to be encouraged to count sets of objects and then probed for how many are in the set. The child needs to remember the last number represents the quanity of the set. Cardinality and quantity are related in counting concepts.
Unitizing: Our number system groups objects into 10 once 9 is reached. We use a base 10 system whereby a 1 will represent ten, one hundred, one thousand etc. Of the counting principles, this one tends to cause the greatest amount of difficulty for children.
I'm sure you'll never look at counting quite the same way when working with your children. More importantly, always keep blocks, counters, coins or buttons to ensure that you are teaching the counting principles concretely. The symbols won't mean anything without the concrete items to back them up.