- Analyze the similarities and differences between the fraction standards in the Common Core State Standards Initiative [CCSSI] (2010) and the fractional knowledge of students constructed over the past 20 years in the Interdisciplinary Research on Number [IRON] project (Author, 2010b).
- Develop a new understanding of standards for fractions in school mathematics based on the fractional knowledge of students.
We agree with the members of the National Mathematics Advisory Panel (2008) that students need to be proficient with fractions, and that fractions are among “the Critical Foundations of Algebra” (p. 46). However, there is good reason for the comment by Davis and colleagues (1993) that, “the learning of fractions is not only very hard, it is, in the broader scheme of things, a dismal failure” (p. 63). Essentially, with the exception of part-whole concepts, fractions are multiplicative concepts, and the involved multiplicative reasoning is based on reasoning with three levels of units. The reasoning that produces a fraction equal to a given fraction as well as improper fractions also involves three levels of units. Furthermore, adding fractions, multiplying fractions, producing classes of equal fractions, and dividing fractions are all based on taking three-levels-of-units structures as given in reasoning. Yet the reasoning that produces three levels of units is not easily engendered even in sixth grade students who can engage in the reasoning that produces two levels of units (Author, 2007a).
An interpretation of what proficiency with fractions might mean is elaborated in the fraction standards in the CCSSI. These standards can be interpreted as indicating the reasoning students could engage in to produce fractional knowledge, while assuming that reasoning to be the same for all students within each grade level. Homogenizing standards in this way is unfortunate because we have found that third grade students who can take three-levels-of-units structures as given in reasoning can learn to reason in most of the ways indicated by the fifth grade standards (Author, 2010b; Olive, 1999). Inversely, we have found that fifth grade students who have constructed only one level of units cannot learn to reason in most of the ways indicated by the third grade standards. In fact, approximately 50% of children entering the third grade in the U.S. have constructed only one level of units, and 50% have constructed two levels of units. By the fifth grade, optimistically, 25% will have constructed three levels of units, 40% two levels of units, and 35% only one level of units. (Author, 1988, 2007b, 2009b; Piaget, Inhelder & Szeminska, 1960). These findings have serious implications for what we should take as standards for fractions in school mathematics.
The symposium will consist of four 15-minute presentations followed by a 30-minute discussion led by a discussant.
The first presenter will focus on students who have constructed one and two levels of units, how students use these levels of units in partitioning operations, and, in turn, how students use these operations in the construction of fraction schemes. The fundamental differences in the partitioning operations between the two levels had profound consequences for the kind of fraction schemes that children constructed in a longitudinal three-year teaching experiment spanning the third through fifth grades (Author, 2010b). The presenter will explain the partitioning operations and the fraction schemes that follow on from them and contrast them with the third and fourth grade fraction standards of the CCSSI.
The second presenter will focus on Common Core Standard 4NF4 and its implications for instruction. This standard calls on teachers to support student conceptions of fractions as iterations of a unit fraction (e.g., 3/5 as three iterations of 1/5). While a worthy goal for improving the quality of U.S. textbooks, which focus narrowly on part-whole conceptions (Watanabe, 2007), the standard must address the cognitive challenges associated with it. Using data from a teaching experiment with a sixth grade student, the presenter will highlight some of these challenges, which include the need for students to coordinate units at two and three levels.
The third presenter will focus on students who have constructed two and three levels of units. Although students at both levels can engage in significant activity with fractions, students at two levels of units are constrained from constructing schemes that produce improper fractions (Author, 2007a), a general multiplying scheme for fractions (Author, 2009a, 2010b), and fractions as quotients (Author, 2010a). Students at three levels of units can construct such schemes given appropriate supportive environments. However, even these students are challenged to construct fractions as operators. Similarities and differences between these findings and the fifth and sixth grade fraction standards of the CCSSI will be discussed.
The fourth presenter will address this culminating question: Given the overview of children’s fractional knowledge in the first three presentations, how could standards be organized to respect and foster this knowledge? The presenter will suggest that standards be written to address competency goals that are realizable for students at different levels of units, regardless of grade level. In addition, comments about grade level expectations should be tied to research findings about students’ levels of units at that grade. For example, in third grade approximately one-half of the students have constructed only one level of units, which indicates where many third grade students would be located on the continuum of competency goals.
The discussant will then lead a discussion centered on the following questions:
1. What should competency-based standards for fractions include and how might they be organized?
2. How might competency-based standards influence approaches to the teaching of fractions?
3. How might competency-based standards impact teacher preparation and professional development?
4. The operations that produce one level of units are thought to be primarily a product of spontaneous development (von Glasersfeld, 1981). Given the crucial role that reasoning with two and three levels of units plays in the construction of fractional knowledge, are the operations that produce two and three levels of units also products of spontaneous development, or can they be engendered in the context of mathematical interaction?
Students’ fraction knowledge requires multiplicative reasoning at three levels of units, a skill that cannot be assumed for most fifth graders. This finding and others contrast with the CCSS Initiative's fraction standards. How could fractions standards be organized to respect and foster students’ fraction knowledge?
Session Type: Research Symposium