**FEATURES** |

Connections Between Generalizing and Justifying: Students' Reasoning with Linear Relationships
*Amy B. Ellis* Research investigating algebra students' abilities to generalize and justify suggests that they experience difficulty in creating and using appropriate generalizations and proofs. Although the field has documented students’ errors, less is known about what students do understand to be general and convincing. This study examines the ways in which seven middle school students generalized and justified while exploring linear functions. Students' generalizations and proof schemes were identified and categorized in order to establish connections between types of generalizations and types of justifications. These connections led to the identification of four mechanisms for change that supported students' engagement in increasingly sophisticated forms of algebraic reasoning: (a) iterative action/reflection cycles, (b) mathematical focus, (c) generalizations that promote deductive reasoning, and (d) influence of deductive reasoning on generalizing. |

Mathematics Coursework Regulates Growth in Mathematics Achievement
*Xin Ma, Jesse L.M. Wilkins* Using data from the Longitudinal Study of American Youth (LSAY), we examined the extent to which students mathematics coursework regulates (influences) the rate of growth in mathematics achievement during middle and high school. Graphical analysis showed that students who started middle school with higher achievement took individual mathematics courses earlier than those with lower achievement. Immediate improvement in mathematics achievement was observed right after taking particular mathematics courses (regular mathematics, prealgebra, algebra I, trigonometry, and calculus). Statistical analysis showed that all mathematics courses added significantly to growth in mathematics achievement, although this added growth varied significantly across students. Regular mathematics courses demonstrated the least regulating power, whereas advanced mathematics courses (trigonometry, precalculus, and calculus) demonstrated the greatest regulating power. Regular mathematics, prealgebra, algebra I, geometry, and trigonometry were important to growth in mathematics achievement even after adjusting for more advanced courses taken later in the sequence of students' mathematics coursework. |

Professional Development Focused on Children's Algebraic Reasoning in Elementary School
*Victoria R. Jacobs, Megan Loef Franke, Thomas P. Carpenter, Linda Levi, Dan Battey* A yearlong experimental study showed positive effects of a professional development project that involved 19 urban elementary schools, 180 teachers, and 3735 students from one of the lowest performing school districts in California. Algebraic reasoning as generalized arithmetic and the study of relations was used as the centerpiece for work with teachers in Grades 1–5. Participating teachers generated a wider variety of student strategies, including more strategies that reflected the use of relational thinking, than did nonparticipating teachers. Students in participating classes showed significantly better understanding of the equal sign and used significantly more strategies reflecting relational thinking during interviews than did students in classes of nonparticipating teachers. |

Proof and Proving in School Mathematics
*Andreas L. Stylianides* Many researchers and curriculum frameworks recommend that the concept of proof and the corresponding activity of proving become part of students' mathematical experiences throughout the grades. Yet it is still unclear what "proof" means in school mathematics, especially in the elementary grades, and what role the teacher has in cultivating proof and proving among their students. In this article, I propose a conceptualization of the meaning of proof in school mathematics and use classroom episodes from third grade to elaborate elements of this conceptualization and to illustrate its applicability even in the early elementary grades. Furthermore, I use the conceptualization to develop a tool to analyze the classroom episodes and to examine aspects of the teachers' role in managing their students’ proving activity. This analysis supports the development of a framework about instructional practices for cultivating proof and proving in school mathematics. |