Educational Studies in Mathematics

Empson, Susan; Junk, Debra; Dominguez, Higinio; Turner, Erin

doi: 10.1007/s10649-005-9000-6pmid: N/A

Although equal sharing problems appear to support the development of fractions as multiplicative structures, very little work has examined how children's informal solutions reflect this possibility. The primary goal of this study was to analyze children's coordination of two quantities (number of people sharing and number of things being shared) in their solutions to equal sharing problems and to see to what extent this coordination was multiplicative. A secondary goal was to document children's solutions for equal sharing problems in which the quantities had a common factor (other than 1). Data consisted of problem-solving interviews with students in 1st, 3rd, 4th, and 5th grades (n=112). We found two major categories of strategies: (a) Parts Quantities strategies and (b) Ratio Quantities strategies. Parts quantities involved children's partitions of continuous units expressed in terms of the number of pieces that would be created. Ratio quantities involved children's creation of associated sets of discrete quantities. Within these strategies, children drew upon a range of relationships among fractions, ratio, multiplication, and division to mentally or physically manipulate quantities of sharers and things to produce exhaustive and equal partitions of the items. Additionally, we observed that problems that included number combinations with common factors elicited a wider range of whole-number knowledge and operations in children's strategies and therefore appeared to support richer interconnections than problems with relatively prime or more basic number combinations.

Crespo, Sandra

doi: 10.1007/s10649-005-9006-0pmid: N/A

In this article, the author examines the character of the conversations generated in an elementary teacher group as they worked on mathematical problems together and analyzed their students' work. Two distinct forms of talk — exploratory and expository — were found. The first type of talk occurred most prominently when discussions centered on the teachers' own mathematical work and the second type when conversations centered on that of their students. By examining closely the few occasions when the groups' expository talk turned exploratory, the author explores how both the nature of the tasks and the range and type of facilitator conversational strategies can play significant roles in promoting and interrupting these conversational patterns to educational ends.

Lloyd, Gwendolyn

doi: 10.1007/s10649-005-9004-2pmid: N/A

This report describes four preservice secondary teachers' fictional accounts about mathematics classrooms. The narrative structures and meanings of the preservice teachers' stories, written while the preservice teachers were enrolled in a secondary methods course, were examined with the goal of gaining insight into the preservice teachers' emerging identities as mathematics teachers. Although past research has examined experienced teachers' stories, little research, if any, has focused on stories written by preservice mathematics teachers or on intentionally fictional stories about mathematics classrooms. This report uses methods of narrative analysis to develop understandings about preservice teachers' storied identities and specifies areas for further investigation into the role of narrative in mathematics teacher education.

Zbiek, Rose; Conner, Annamarie

doi: 10.1007/s10649-005-9002-4pmid: N/A

Views of mathematical modeling in empirical, expository, and curricular references typically capture a relationship between real-world phenomena and mathematical ideas from the perspective that competence in mathematical modeling is a clear goal of the mathematics curriculum. However, we work within a curricular context in which mathematical modeling is treated more as a venue for learning other mathematics than as an instructional goal in its own right. From this perspective, we are compelled to ask how learning of mathematics beyond modeling may occur as students generate and validate mathematical models. We consider a diagrammatic model of mathematical modeling as a process that allows us to identify how mathematical understandings may develop or surface while learners engage in modeling tasks. Through examples from prospective teachers' mathematical modeling work, we illustrate how our diagrammatic model serves as a tool to unpack the intricacies of students’ mathematical experience while engaging in modeling tasks.