**FEATURES** |

Supporting Students' Mathematical Inquiries Through Reading
*Marjorie Siegal, Raffaella Borasi, Judith Fonzi* The purpose of this article is to identify specific functions that reading, in combination with writing and talking, can serve in mathematical inquiries and thus to contribute to a better understanding of how inquiry experiences can be planned and supported in mathematics classrooms. This purpose is achieved through an analysis of 3 classroom experiences in which secondary mathematics students engaged in "inquiry cycles" on quite different topics. These instructional experiences were developed by a collaborative team of mathematics teachers, mathematics education researchers, and a reading researcher in the context of action research and teacher research. Analysis of the data led to the identification of 30 functions of reading that are specific to distinct elements of an inquiry cycle. On the basis of these findings we suggest that reading can serve multiple roles in inquiry-based mathematics classes and, in doing so, can afford students unique opportunities for learning mathematics. |

Using Concept Maps to Assess Conceptual Knowledge of Function
*Carol G. Williams* In this study I examine the value of concept maps as instruments for assessment of conceptual understanding, using the maps to compare the knowledge of function that students enrolled in university calculus classes hold. Twenty-eight students, half from nontraditional sections and half from traditional sections, participated in the study. Eight professors with PhDs in mathematics also completed concept maps. These expert maps are compared with the student maps. Qualitative analysis of the maps reveals differences between the student and expert groups as well as between the 2 student groups. Concept maps proved to be a useful device for assessing conceptual understanding. |

Use of Multiplicative Commutativity by School Children and Street Sellers
*Analucia D. Schliemann, Claudia Araujo, Maria Angela CassundiSuzana Macedo, Lenice Nicias* We analyzed use of the commutative property for solving multiplication problems by children who learn about multiplication in schools and by street vendors who solve multiplication problems through repeated addition. Subjects were Brazilian street vendors with irregular school attendance, who had received no, or very little, school instruction on multiplication, and 1st- to 3rd-grade Brazilian school children. Results from 2 studies show that use of commutativity to solve multiplication problems is closely related to use of multiplication. Street sellers who rely exclusively on repeated addition to solve such problems may, however, use multiplicative commutativity if it represents a clear reduction in the number of computational steps needed to reach a solution. But their justifications for its use are often based on knowledge about multiplication. |

The Developmental Nature of Ability to Solve One-Step Word Problems
*Constantinos Christou, George Philippou* Recently many researchers have focused on analyzing the structure of one-step word problems and the solution strategies employed by pupils. We have investigated the effect of mental schemes corresponding to additive and multiplicative situations in the process of interpreting and solving problems. The relative difficulties of problems classified according to their situations is considered through a written test administered to pupils in Grades 2, 3, and 4. The results seem to support the assumption that there is a developmental pattern in pupils' thinking, depending on the problem situation. |

The Empty Number Line in Dutch Second Grades: Realistic Versus Gradual Program Design
*Anton S. Klein, Meindert Beishuizen, Adri Treffers* In this study we compare 2 experimental programs for teaching mental addition and subtraction in the Dutch 2nd grade (N = 275). The goal of both programs is greater flexibility in mental arithmetic through use of the empty number line as a new mental model. The programs differ in instructional design to enable comparison of 2 contrasting instructional concepts. The Realistic Program Design (RPD) stimulates flexible use of solution procedures from the beginning by using realistic context problems. The Gradual Program Design (GPD) has as its purpose a gradual increase of knowledge through initial emphasis on procedural computation followed by flexible problem solving. We found that whereas RPD pupils showed a more varied use of solution procedures than the GPD pupils, this variation did not influence the procedural competence of the pupils. The empty number line appears to be a very powerful model for the learning of addition and subtraction up to 100. |

The Value (and Convergence) of Practices Suggested by Motivation Research and Promoted by Mathematics Education Reformers
*Deborah Stipek, Julie M. Salmon, Karen Givvin, Elham Kazemi, Geoffrey Saxe, Valanne L. MacGyvers* In this study we discuss convergence between instructional practices suggested by research on achievement motivation and practices promoted in the mathematics instruction reform literature, and we assess associations among instructional practices, motivation, and learning of fractions. Participants included 624 fourth- through sixth-grade students and their 24 teachers. Results indicated that the instructional practices suggested in literature in both research areas positively affected students' motivation (e.g., focus on learning and understanding; positive emotions, such as pride in accomplishments; enjoyment) and conceptual learning related to fractions. Positive student motivation was associated with increased skills related to fractions. |