The Components of Number Sense: An Instructional Model for Teachers

Article excerpt

In recent years much attention has been placed on the relatively poor math performance of students in the United States (Gonzalez et al., 2004; Lemke et al., 2004; National Center for Education Statistics, 1999; National Research Council, 2001). Increased attention has also been paid to the struggling learner and mathematics. This includes issues regarding assessment (Gersten, Clarke, & Jordan, 2007); low-performing students in reformbased classrooms (Baxter, Woodward, & Olson, 2001); and general recommendations for the struggling student by the National Math Panel (Gersten et al., 2008).

The mathematical knowledge of teachers has also been investigated, and student success has been tied to the subtle factors of teacher implementation choices regarding problem sets, questioning techniques, and math connections (Hiebert & Stigler, 2000; Hill, Rowan, & Ball, 2005; Stigler & Hiebert, 2004). Strong teacher implementation choices appear to be influenced by teacher knowledge and flexibility with the mathematics being taught. Furthermore, it has been demonstrated qualitatively that elementary teachers in the United States tend to lack a "profound understanding" of the fundamentals of the mathematics they teach (Ma, 1999).

Number Sense and Instructional Practice

At the heart of the recent focus on mathematics has been an increased emphasis on developing students' number sense. Ironically, although growing as a force in the education literature, number sense has not been clearly defined for teachers.

Teachers need specific support in understanding how to develop number sense in students, to guide their learning as they plan for and provide instruction (Ball & Cohen, 1996) and, ultimately, to ensure that they are spending time encouraging students to do the thinking that will improve number sense. A focus on content knowledge has been found to be an effective component of professional development for teachers (Garet, Porter, Desimone, Birman, & Suk Yoon, 2001; Hill et al., 2005), and teacher content knowledge in mathematics has an impact on student performance (Hill et al.). In our work with hundreds of teachers throughout our state, we have found it necessary to support teachers with a model for number sense development that, first and foremost, supports a deep understanding of the mathematics itself. Using this model as the framework for the North Carolina Math Foundations training, we have been able to show teacher knowledge growth as measured by the Learning Mathematics for Teaching (LMT) Measures developed at the University of Michigan.

Teachers are increasingly faced with standard course of study documents listing number sense as a goal of instruction (e.g., in Washington, Missouri, North Carolina). These standards tend to present number sense in a perfunctory fashion that does little to delineate for the teacher how students acquire that number sense. Even those who do research to develop our understanding of number sense continue to refer to the phrase "difficult to define but easy to recognize" (Gersten, Jordan, & Flojo, 2005). In 2001, Kalchman, Moss, and Case describe number sense as

The characteristics of good number sense include: (a) fluency in estimating and judging magnitude, (b) ability to recognize unreasonable results, (c) flexibility when mentally computing, (d) ability to move among different representations and to use the most appropriate representations. (p. 2)

But for the teacher, the questions still remain: How do I get my pupils to gain these characteristics? What does this mean about how I should teach mathematics?

In other words, number sense is poorly outlined for the teaching community (if students can solve certain problems, then they have number sense) and is essentially defined in circular terms. This circular tendency perhaps reflects, unwittingly, a cultural vision of mathematical ability as some- thing that is gifted to the individual rather than learned through specific patterns of habit and practice (Dehaene, 1997). …