By the very nature of mathematics, mathematical ideas build one upon the other. In fact, a child's mathematical development closely resembles mathematics' historical development. For example, children first understand the natural (or counting) numbers. They first experience this mathematical idea intuitively in their daily world with hands-on activities, and then by extension move onto more formal symbols. Extensions to even higher-level operations are built on the basic understandings of addition and subtraction. Multiplication is first viewed as "repeated addition." This leads to the understanding that relating the child's world to the world of mathematics gives the child a reason to want to know.
As students become developmentally ready for more advanced levels of abstraction, they can explore more sophisticated mathematical ideas. But early experiences in an informal setting build the foundation for more rigorous development in high school and later. An example of this development is in measurement and scale - from the infinitesimal to the gigantic. First, ideas of "how big" rely on comparison, best learned by using objects that initially may be very different in size, such as a mouse and an elephant. Piaget's "conservation" abilities do not occur developmentally until a certain maturity is reached. For example, a young child will judge a tall, slender glass to be "bigger" than a shorter glass of larger diameter, even if a liquid poured from the shorter glass fills the slender glass to over-flowing. Later, when the child is developmentally ready, volume measurement becomes meaningful.
In order to be effective learning tools for children, instructional materials must be appropriate for them. Recognizing the change in the progressive sophistication of children's logical reasoning as they mature provides opportunity for aligning learning with developmental differences. For example, in the middle grades, most children begin to comprehend "if, then" reasoning, an example of informal deduction. As these children mature and develop intellectually through high school, they begin to understand and use the rules of formal deductive reasoning. It would be inappropriate to present mathematical proofs based on formal rules of deductive reasoning to most middle-grade learners. Similarly, it would be inappropriate not to use some formal logical reasoning in the teaching of all high school students.
The mathematical habits of mind increase in rigor and depth as learners develop through their school-based experiences. The values and capabilities that capture the essence of mathematical "thinkers and doers" in our world must be introduced and strengthened throughout the schooling process. Mathematical thinking and doing must become as natural for children as listening, speaking, reading, and writing. Problem solving, reasoning, connecting and communicating - mathematical habits of mind, must be fluent capabilities for all our children.