Number sense refers to a person's general understanding of number and operations along with the ability to use this understanding in flexible ways to make mathematical judgments and to develop useful strategies for solving complex problems
(Burton, 1993; Reys, 1991). Researchers note that number sense develops gradually, and varies as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms (Howden, 1989).
According to McIntosh et al. (1992),
number sense refers not only to a person’s understanding of number and operations, but also to his/her inclination and ability to use this understanding in flexible ways in order to make sense of situations involving number and quantity.
Numerate people regard numbers as useful and see mathematics as having regularity. A teaching program should reflect this attitude.
Other definition from Gersten &
Chard, 1999, numbers sense is “a child’s fluidity and flexibility
with numbers, the sense of what numbers mean, and an ability to
perform mental mathematics and to look at the world and make
of Number Sense.
While number sense appears to be greater than the sum of its parts, we can identify various aspects of number sense (McIntosh et al., 1992):
Knowledge of and facility with numbers.
Students with number sense…
- recognize orderliness in numbers and regularity in the number system
- know that numbers can be represented in many different ways, some being more efficient than others in certain problem situations
- develop useful mental referents (“benchmarks”) for thinking about numbers have an intuitive sense for the relative sizes of numbers
- work comfortably with different kinds of numbers, including rates, measures, and probabilistic statements
Knowledge of and facility with operations.
Students with number sense…
- understand the operations and their effects on various numbers
- make connections between mathematical properties and practical applications
- recognize operations are related and apply these relationships
- construct algorithms for computation and modify them to fit different situations
Ability to apply the above knowledge and skills to computational situations.
Students with number sense…
- understand and communicate the relationship between the context of the problem and the related computation(s)
- are aware of, can identify and use multiple solution strategies
- are able to use efficient representations of numbers flexibly and proficiently
- approximate numerical answers and judge whether a particular answer to a particular problem is reasonable
- review their work and results
Students acquire number sense gradually, experiencing many informal pre-number activities before they begin their formal schooling. But growing older does not guarantee a student’s continued development of number concepts or skills. In fact, the development of number sense skills does not occur naturally for most students in the current school curriculum. The major stumbling block is the rush to develop students’ use of standard algorithms to perform operations. Students quickly fixate on these methods because they can be “executed without having to think” (McIntosh et al., 1992, p.Ê 3). We need to turn an awareness of number into an understanding of number. We need to move more thoughtfully from a conceptual understanding of operation to the development of informal algorithms. The following chart can be used to make curricular decisions.
the development of Number Sense
The classroom teacher of mathematics can plan for the development of number sense around these four components identified in the
Professional Standards for Teaching Mathematics
I. Worthwhile Tasks
We need to provide students with rich contextual activities which support the development of problem solving as well as the different components of number sense. Calculations should be done for a purpose and the purpose made explicit to students. Calculations within a problem-solving context will help students avoid the routinized, mechanistic approaches that block the development of number sense. When the context is clear to students, calculated answers must be interpreted. (Is this 3.5 cars or miles? Is this 16 miles per gallon or 16 miles per hour?) A clear context can also motivate students to take the time and energy to evaluate the reasonableness of calculated answers. Invented solution strategies and calculation methods should be encouraged long before students are asked to construct formal algorithms.
We need to ask process questions that cause students to reflect on their thinking. We need to provide them with clear visual images of the concepts to hold in their minds. This involves a “quality over quantity” attitude towards problem completion. Our emphasis should be on helping students understand a given problem by looking at it from multiple perspectives and investigating multiple solutions, rather than on attempting as many problems as possible in a given period. Students should be encouraged to generate, share, write about, and make sense of many solution strategies to a given problem. This approach often helps them appreciate the efficiency of standard algorithms (Reys, 1994, p. 117).
In this technological age it is necessary to make use of appropriate calculator tools with students. Students must learn when and how to calculate in various ways including paper and pencil methods, mental approximation and computation, and electronic methods. Students should realize there are times when it is
inappropriate to use a calculator just as there are times when it is
inappropriate not to. Calculators should be seen as powerful learning tools as well as aids to computing.
The development of a student’s number sense is ultimately related to his/her ability to think and make connections. We need to promote the asking of key questions before, during, and after a solution process.
- What information do I know?
- What kind of number is this?
- How is it used?
- What information do I need?
- Does this problem seem to have one right answer or many possible answers?
- Do I need an exact answer or an estimated answer?
- What seems to be a reasonable approach?
- What are some different solution strategies that might work?
- Is there a more efficient way to represent these numbers for this approach?
- Does this answer make sense for this situation?
- How do I interpret this calculation in light of the context of the problem
These “internal questioning tools” are critical even when accompanied by the use of external electronic tools.
A classroom atmosphere that encourages clear conceptual images and that promotes exploration, thinking, shared solutions, and discussion is an important ingredient in students’ development of number sense. The shared interactions of all members of the classroom are necessary to make the connections needed to promote students’ sense of number. Understanding numbers and operations and applying them appropriately and efficiently in problem solving situations develops gradually, but can be promoted by a community of learners collectively attempting to make sense of its work.
Number sense is no easier to evaluate than it is to define. “We are going to have to observe students solving various kinds of problems involving number, and then make judgments about the extent to which students seem to be reasoning effectively about number” (Resnick, p. 36-37). We will need rich problem situations, multiple assessment tools, and informed observations to make decisions about student growth in concepts and skills.