Many calls for reform in school mathematics emphasize number sense (Australian Education Council, 1991; Cockroft, 1982; Japanese Ministry of Education, 1989; National Council of Teachers of Mathematics, 1989; 2000; Swedish Ministry of Education and Science, 1994). But what is number sense? According to Case (1998), "Number sense is difficult to define, but easy to recognize. Students with good number sense can move seamlessly between the real world of quantities and the mathematical world of numbers and numerical expressions. They can invent their own procedures ... represent the same number in multiple ways.... recognize benchmark numbers and number patterns.... and gross numerical errors...." (p. 1).
I am using number sense to refer to "the general understanding of number and operations, along with the ability and inclination to use this understanding in flexible ways to make mathematical judgments and to develop useful and efficient strategies for managing numerical situations" (Reys & Reys, McIntosh, Emanuelsson & Johansson, and Der, 1999, p. 61).
Most research on number sense has, focused on "average" children's number sense (e.g. Reys & Reys et al., 1999; Turner, 1996; Yang, 2002), while some studies have focused on the number sense of children with learning disabilities (e.g. Gersten & Chard, 1999; Griffin, Case, and Siegler, 1994). However, studies on teachers' number sense (e.g. Ma, 1999; Menon, 1999) have been limited, and hence this study serves to augment knowledge about teachers' number sense, specifically preservice teachers' number sense.
Participants and Methodology
This study involved 142 preservice teachers from four groups of mathematics methods classes/sections I taught. These students were in a teaching credential program, which included 8 credit hours of mathematics education, leading to a K-8 teaching certification. Since these preservice teachers were going to teach mathematics that can be considered foundational to students in K-8, I believed it would be beneficial for me to find out these preservice teachers' mathematical competence, specifically their own number sense.
Many of them (almost 90%) had not taken algebra and geometry in high school. But all of them had taken mandatory mathematics content courses for teachers (that included algebra and geometry), taught by the mathematics department faculty of the university. (In this university, the math methods courses are taught by the college of education faculty, but the math content courses are taught by math department faculty.)
In spite of their taking, and passing, the mandatory university math content courses, results from an informal survey revealed that about 85% of them still lacked confidence in teaching middle school mathematics, mainly because they were unsure of their ability to successfully do mathematics content at that level.
On the first day of their mathematics education class, these 142 preservice students were given a 10 item, multiple choice paper and pencil test on number sense. For each item, they were given space on the test paper itself, to give written explanations for their answers. They were given 25 minutes to complete the test.
The 10 items were selected from those used in previous studies on number sense (e. g. Menon, 1999; Reys & Reys et al., 1999). The components of number sense (also based on previous studies), together with the relevant item number assigned to test them, are given next:
1. To make mathematical judgments (J), for example, by determining appropriateness and sufficiency of information--item #s 1 & 2.
2. To develop useful and efficient strategies for managing numerical situations (E), for example, by using estimation and number relationship--item #s 3 to 5.
3. The general understanding of number and operations (U), especially those related to fractions and decimals--item #s 6 to 10. …