Keeping track of time has intrigued people throughout history. The constant urge to harness time has resulted in many attempts to perfect the calendar. The study of calendars offers students many opportunities to investigate measurement issues associated with time, the revolution of the earth around the sun, and the historical development of the calendar as civilization became more dependent on keeping accurate time.

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### Vanessa R. Pitts Bannister and Jesse L. M. Wilkins

In *Principles and Standards for School Mathematics* (NCTM 2000), understandings of patterns, relations, functions, mathematical models, and quantitative relationships are recognized as key facets of algebraic thinking. In essence, algebraic thinking “embodies the construction and representation of patterns of regularities, deliberate generalization, and most important, active exploration and conjecture” (Chambers 1994, p. 85). Algebraic thinking should function as a means of shifting from arithmetic concepts to algebraic concepts (Chappell 1997). This shift would have occurred if there exists reasoning about relationships between quantities, rather than the specific quantities themselves (Ferrini-Mundy, Lappan, and Phillips 1997; Yackel 1997). Research shows that this arithmetic to algebraic shift is difficult for students (Stacey and MacGregor 2000). Therefore, it is imperative to explore students' reasoning as they approach problems that elicit algebraic thinking. For this reason, we will present and discuss samples of student work regarding problems that promote algebraic thinking.

### Xin Ma and 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 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.

### Jesse L. M. Wilkins and David Hicks

As technological advances continue to help more people make connections with the entire world, students must understand how to use and interpret information shown in different maps of the world (Geography Education Standards Project 1994; Freese 1997). However, mental-mapping research suggests that students in the United States have major misconceptions about proportions, locations, and perspective when they work with maps (Dulli and Goodman 1994; Stoltman 1991).

### Jesse L. M. Wilkins and Anderson Norton

Teaching experiments have generated several hypotheses concerning the construction of fraction schemes and operations and relationships among them. In particular, researchers have hypothesized that children's construction of splitting operations is crucial to their construction of more advanced fractions concepts (Steffe, 2002). The authors propose that splitting constitutes a psychological structure similar to that of a mathematical group (Piaget, 1970b): a structure that introduces mutual reversibility of students' partitioning and iterating operations that the authors refer to as the splitting loope. Data consisted of 66 sixth–grade students' written performance on 20 tasks designed to provoke responses that would indicate particular fractions schemes and operations. Findings are consistent with hypotheses from related teaching experiments. In particular, they demonstrate–consistent with the notion of the splitting loope—that equipartitioning and the partitive unit fraction scheme mediate the construction of splitting from partitioning and iterating operations.

### Anderson Norton and Jesse L. M. Wilkins

Piagetian theory describes mathematical development as the construction and organization of mental operations within psychological structures. Research on student learning has identified the vital roles of two particular operations–splitting and units coordination–play in students' development of advanced fractions knowledge. Whereas Steffe and colleagues (e.g., Steffe, 2001; Steffe & Olive, 2010) describe these knowledge structures in terms of fractions schemes, Piaget introduced the possibility of modeling students' psychological structures with formal mathematical structures, such as algebraic groups. This paper demonstrates the utility of modeling students' development with a structure that is isomorphic to the positive rational numbers under multiplication–the splitting group. We use a quantitative analysis of written assessments from 58 eighth grade students to test hypotheses related to this development. Results affirm and refine an existing hypothetical learning trajectory for students' constructions of advanced fractions schemes by demonstrating that splitting is a necessary precursor to students' constructions of 3 levels of units coordination.

### Michelle Muller Wilkins, Jesse L. M. Wilkins, and Tamra Oliver

Description of the Mathematics Investigation Center (MIC), a tool to help elementary teachers differentiate the curriculum for their gifted mathematics students. Using the same mathematical theme that the rest of the class is studying, the activities provide depth for the gifted students by shifting from a computation level to a problem solving level.

### Anderson Norton, Jesse L. M. Wilkins, Michael A. Evans, Kirby Deater-Deckard, Osman Balci, and Mido Chang

Explore a new app that allows students to develop a more sophisticated understanding of fractions.

### Anderson Norton, Jesse L. M. Wilkins, and Cong ze Xu

Through their work on the Fractions Project, Steffe and Olive (2010) identified a progression of fraction schemes that describes students' development toward more and more sophisticated ways of operating with fractions. Although several quantitative studies have affirmed this progression, the question has remained open as to whether it is specific to the U.S. classrooms in which these studies were conducted or whether it describes a developmental progression that crosses international boundaries. The purpose of our replication study was to address that question using data gathered from written assessments of 76 5th- and 6th-grade students in China. Results indicate a remarkably similar progression among students in the United States and students in China.