In the infant stage, between the ages of zero and 1, the baby is developing the skills necessary for later mathematical comprehension. According to Tufts University Assistant Clinical Pediatric Professor Leo Leonidas, she is mostly an observer at this point. She is absorbing information.
By the time the child reaches the age of 2, he can recite the names of numbers. He can count to five, and also point to an object in the room. As Leonidas notes, the parent can use the instance of pointing as a teachable moment in the child's process of learning to count. He can encourage the child to count a group of objects, such as pennies, and point to them, and have him call out the numbers.
By age 4, the child have acquired the concept of quantity, according to Leonidas. The child can now associated the name of the number with the act of counting. She can point at objects and comfortably tell how many, real or imaginary, are present.
By kindergarten, he is able to point, count, add and subtract using manipulatives, such as blocks, according to the Victorian Essential Learning Standards website. As Leonidas suggests, parents can help children at this stage by using tangible objects. This act will reinforce the teaching he is receiving at school.
According to the Victorian Essential Learning Standards website, by age 8, she increases in the ability to understand both the concrete and the abstract. She is able to make and expand upon hypotheses and can use abstractions in number, space, measurement, probability and data. She can demonstrate such knowledge by the use of graphs and technology, such as a calculator.
By age 10, the student can relate math concepts to real life. He may evidence this, for instance, by being able to solve simple story problems relating to a trip to the grocery store which require him to subtract. At this stage the student may develop the skill of building larger abstractions on the ones he's already mastered, leading to a future time when she will excel at advanced algebra, geometry, trigonometry and calculus.