The Role of JRME in Advancing Learning and Teaching Elementary School Mathematics
Battista, Michael T., Larson, Carol Novillis, Teaching Children Mathematics
The Journal for Research in Mathematics Education (JRME) is the research journal of the National Council of Teachers of Mathematics. It is a "forum for disciplined inquiry into the learning and teaching of mathematics at all levels--from preschool through adult" (JRME Editorial Board 1993, 3). JRME's twenty-fifth anniversary year is 1994. As part of the celebration of this event, members of JRME's editorial panel have written articles for the Council's other journals that describe the role that JRME and research have played in promoting teaching and learning mathematics. This article will focus on JRME's contribution to the view of learning and teaching elementary school mathematics embodied in current curricular recommendations for school mathematics.
Research on Children's Learning
Research in JRME illustrates how prevailing theories about how students learn mathematics have moved away from a behaviorist tradition, which focuses on what students do, to a more constructivist view, which focuses on how students think. According to a constructivist view, students learn mathematics meaningfully as they personally con-street mental structures and operations that enable them to deal with problematic situations, organize their ideas about the world, and make sense of their interactions with others. The constructive process occurs as students reflect on and abstract the mental and physical actions they perform on their current representations of the world (see, e.g., Clement ; Cobb et al. ; Cobb et al. ).
In a constructivist approach to teaching, the goal is to guide students in building mathematical structures that are more complex, abstract, and powerful than those they already possess. Such guidance can occur only if it is based on a knowledge of what conceptual structures the students might make and how the students might change during the course of instruction (Cobb et al. 1991). Along this line, JRME has reported much research that has investigated the cognitive structures and operations that students bring to bear on problems in elementary school mathematics. In this article, several important themes that recur in the research will be described, an example of an essential concept in mathematics whose in-depth research analysis has important implications for curriculum redesign will be given, and some specific classroom applications of the research will be listed.
* Students often apply informal, common-sense knowledge rather than formal, school-taught procedures when solving mathematical problems. Sometimes this informal knowledge is powerful enough to solve the problems; sometimes it is not. For instance, students in grades 1-3 often solve addition and subtraction word problems by employing informally learned counting strategies to model the actions described in the problems (Carpenter and Moser 1984; De Corte and Verschaffel 1987). Similarly, for a pictorially presented problem, Lamon (1993) asked sixth graders if 19 food pellets would be enough for 9 aliens, given that 3 aliens need 5 pellets. She found that rather than comparing ratios, most students modeled the problem by matching 3 aliens with 5 pellets three times, correctly determining that 4 pellets were left over. However, both Lamon (1993) and Mack (1990) found that many of the students' informal procedures were inadequate for solving more complex fraction and ratio problems. Moreover, Clements and Battista (1990) found that fourth graders have many informal, everyday conceptualizations of geometric ideas that actually compete with formal concepts. For instance, students often think of angles as tilted lines, straight as meaning perpendicular, and a rectangle as having two long sides and two short sides, which means that for them a square is not a rectangle.
* Students' informal, intuitive ideas are, for the most part, unconnected to the formal concepts, procedures, and symbols they learn in school. …