Thus, learning how to add and subtract multidigit numbers does not have to involve entirely new and unrelated ideas. The same observation can be made for multiplication and division.

Conceptual understanding helps students avoid many critical errors in solving problems, particularly errors of magnitude. For example, if they are multiplying 9.83 and 7.65 and get 7519.95 for the answer, they can immediately decide that it cannot be right. They know that 10×8 is only 80, so multiplying two numbers less than 10 and 8 must give a product less than 80. They might then suspect that the decimal point is incorrectly placed and check that possibility.

Conceptual understanding frequently results in students having less to learn because they can see the deeper similarities between superficially unrelated situations. Their understanding has been encapsulated into compact clusters of interrelated facts and principles. The contents of a given cluster may be summarized by a short sentence or phrase like “properties of multiplication,” which is sufficient for use in many situations. If necessary, however, the cluster can be unpacked if the student needs to explain a principle, wants to reflect on a concept, or is learning new ideas. Often, the structure of students’ understanding is hierarchical, with simpler clusters of ideas packed into larger, more complex ones. A good example of a knowledge cluster for mathematically proficient older students is the number line. In one easily visualized picture, the student can grasp relations between all the number systems described in chapter 3, along with geometric interpretations for the operations of arithmetic. It connects arithmetic to geometry and later in schooling serves as a link to more advanced mathematics.

As an example of how a knowledge cluster can make learning easier, consider the cluster students might develop for adding whole numbers. If students understand that addition is commutative (e.g., 3+5=5+3), their learning of basic addition combinations is reduced by almost half. By exploiting their knowledge of other relationships such as that between the doubles (e.g., 5+5 and 6+6) and other sums, they can reduce still further the number of addition combinations they need to learn. Because young children tend to learn the doubles fairly early, they can use them to produce closely related sums.10 For example, they may see that 6+7 is just one more than 6+6. These relations make it easier for students to learn the new addition combinations because they are generating new knowledge rather than relying on rote memorization. Conceptual understanding, therefore, is a wise investment that pays off for students in many ways.

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