Meaning and Importance

Our students often see too little relationship between what is happening in their mathematics classrooms and their daily lives. It is essential that the mathematics curriculum build on the sense of number that students bring to school. Problem solving offers the major avenue to that link and to the development of number sense. Problems and numbers which arise in the context of the students world are more meaningful than traditional textbook exercises and help to promote the relationship between the operations and the numbers involved. Frequent use of estimation and mental computation are also important ingredients in the development of number sense as are regular opportunities for student communication. The discussion of their own invented strategies for problem solutions helps to strengthen the students intuitive understanding of numbers and number relations.

Number sense is not inherent in a person's ability to perform numerical computations. A "sense-building mode" is best established when students are provided with opportunities to explore number relationships, are encouraged to challenge and question, and are allowed to experiment to discover strategies and techniques of their own that ease the path to the solution of mathematical problems.

K-12 Development and Emphases

The development of personal meaning for numbers should be reinforced in the middle grades with an extension to other numbers and notations such as percents and roots. Student experiences should include exploration of the properties of sets of numbers. High school students should extend their meaning of number to the real number system and a recognition that still other number systems exist. They should have the opportunity to develop intuitive proofs of properties of operations and sets of numbers such as closure, commutativity, and associativity.

Students must also develop a facility for working with different forms of numbers. To intelligently select the right form of numbers to use for a particular task, a student must be very comfortable with order and comparison relations and with various approaches to establishing equivalence among the variety of types of numbers we use regularly in our society. Statements about one particular quantity might best be expressed with a fraction, a percent, a decimal, a ratio, an approximated whole number, or some other number form. The correct choice depends on the context of the use of the number, but must be built upon an adequate understanding of each form and the interrelationships among them.

One empowering way to achieve these understandings in the classroom is through the identification and description of number patterns and the use of pattern-based thinking. For example, the examination and modeling of many pairs of fractions with equal numerators help students develop the generalization that the fractions with larger denominators represent smaller quantities. Activities promoting pattern-based thinking can assist students in making similar generalizations about other number forms and their relationships as well as to build initial notions of yet other types of important number concepts such as odd and even, prime numbers, factors and multiples.

Graduates of our schools also must be able to wisely use numbers wherever they are encountered in real life. They must develop an awareness of numbers and their uses. Numbers are used as counts, as measures, as labels, and as locations and each use has unique restrictions on appropriate forms and operations. The opportunity to develop the needed familiarity with all of these uses comes through the regular presentation of problem situations which emphasize them. Some activities should focus on the explicit uses of the numbers themselves, however. An elementary grades discussion of why it makes no sense to add the numbers on a town's roadway welcome sign which lists the population, elevation, and year of founding would help challenge thoughtless computational manipulation of numbers as well as establish the standard number uses.

In summary, the commitment to develop number sense requires a paradigm shift in the way students learn mathematics. Our students will only develop strong number sense to the extent that their teachers use pedagogy which encourages the understanding of mathematics as opposed to the memorization of rules and mechanical application of algorithms. Every child has the capability to succeed as a user of mathematics, but the degree of success is directly related to the strength of their number sense. The way to assure that all students acquire a good sense of number is to have them consistently engage in activities which require them to think about numbers and number relationships and to make the connection with quantitative information encountered in their daily lives.