I am going to give you 2 different pathways for different approaches to learning arithmetic. I will use 2 different colors to separate the approaches. Neither approach is better or worse than the other. They are just different and both will be covered by following one pathway then coming back and doing the secondary pathway. Or you could simultaneously use both approaches.
The first approach is number fact mastery by counting numerals. When many people think about mastering numbers facts they fear the dreaded flash cards, rote memorization and endless drills. However there are some simple strategies that can be used to teach fact mastery that are fun and enjoyable.
Although this is the fastest and most direct way to master number facts there is also a downside. A child might well learn that 6+3=9, however they may not have a concrete idea of what this really means.
The alternative pathway to reach the goal of fact mastery relies heavily on manipulatives. This is the most common way that arithmetic is taught to children in the early school years. It can involve a lot of play and group work, which often equals a lot of fun, and better learning outcomes.
However I have found that there are several concepts that very young children struggle with. These included bijection and cardinality. Although some children seem to grasp these concepts easily and at a young age the majority of children need a lot of practice and time.
My recommendation is to consider the age and the ability of the child. Pre-school and Kindergarten aged children will benefit mostly from the second pathway because it strengthens their number sense.
Now that a child is able to count by reading numerals proficiently to 20 both forwards and backwards we are going to move onto the next step.
This is an essential step that separates reciting numbers or reading numbers type of counting from counting an actual amount.
Bijection and Cardinality
Bijection, also known as one-to-one correspondence is the ability to point at an individual object and know that the object represents 1 counting word.
For a child that has been using a number line you can easily check their proficiency by removing the numerals written on a number line and replacing them with a dash or preferably a dot. If a child is struggling you can remove one or two numbers at a time and have them continue reading some of the numbers as they count up and down.
When using manipulatives (objects), while counting it is best to lay out small amounts of objects. Start with 3, and then 5, and then 10. Only move to larger amounts when there is mastery. Have the child first practice touching each object and saying their corresponding number words. Try and encourage them not to point to an object twice or skip an object. You may need to do the pointing yourself, or guide their fingers. You can also practice bijection by moving the objects from one side the table to the other. Or have them line up the objects.
If the child is still struggling have them start with a group of small objects and have them move the objects one by one to a corresponding numeral on a number line. Make sure this is done in order and no numeral is missed.
Cardinality is the ability to recognize that the last number counted is the answer. Or when using manipulatives it is the total amount.
It sometimes takes children a while to stop at the end of a counting sequence because they are still reciting numbers. Each time you count, either using a number line or objects, make sure to ask your child what the last number they counted was. If they are unsure, as is common, tell them. Explain that the last number is like the destination and it is important to stop there and take note of what it is.