Concrete: "What you see is what there is."

1. Children accept the way things look as definitive, for example thinking a short fat glass is bigger than a skinny tall one which is actually the same size. Counting is by rote as is addition. Figuring out 5 + 4 = depends on using fingers on both hands.
2. Children use real objects to understand numbers, but can also "count on" - when adding 5 + 4, they can start at 5 then add up 6, 7, 8, 9. This stage can appear anywhere from age 4-8.
3. Children can identify numbers quickly and easily but still use objects - they see 5 + 4 and know the sum is 9. This stage can develop anywhere from age 6-12
4. Children can use mathematical symbols with ease (=, +, x). Age 6-12 is the range in which this stage can develop.
5. Children can think abstractly. They can examine the consequences of various alternatives, play with ideas, deal with "if this, then what" problems, formulas and equations.