Spatiotemporal Query Algebra Based on Native XML

A formal algebra is essential for applying standard database-style query optimization to XML queries. We propose a spatiotemporal XML data model and develop such an algebra based on Native XML, for manipulating spatiotemporal XML data. After studying NXD spatiotemporal database and query framework, formal representation of spatiotemporal query algebra is investigated, containing logical structure of spatiotemporal database, data type system, and querying operations. It shows that the model and algebra lay a firm foundation for managing spatiotemporal XML data.

Leveraging Dynamic and Dependable Spreadsheets Focusing on Algebraic Thinking and Reasoning

Algebraic thinking and reasoning are important skills for students to develop as they transition from arithmetic to algebra. The Common Core State Standards for Mathematics links these ideas through their mathematical practices where students make sense of mathematical problems, reason abstractly, model with mathematics, and use appropriate tools that engage them in algebraic thinking and reasoning. This chapter focuses on how a specific tool—the spreadsheet—supports students in building their understandings of spreadsheet features and capabilities while also engaging them in thinking more logically about numbers as needed in preparation for formal algebra. The challenge for teachers is to scaffold students' learning about and with spreadsheets as mathematics learning tools for exploring patterns, variables, and linear problems. This chapter describes this transition through multiple problems where students learn to design dynamic and dependable spreadsheets in ways that concurrently develop their algebraic reasoning and thinking through middle grades mathematics problems.

On the Iwasawa theory of the Lubin–Tate moduli space

We study the affine formal algebra$R$of the Lubin–Tate deformation space as a module over two different rings. One is the completed group ring of the automorphism group$\Gamma $of the formal module of the deformation problem, the other one is the spherical Hecke algebra of a general linear group. In the most basic case of height two and ground field$\mathbb {Q}_p$, our structure results include a flatness assertion for$R$over the spherical Hecke algebra and allow us to compute the continuous (co)homology of$\Gamma $with coefficients in $R$.

Developing Meaning for Algebraic Symbols: Possibilities and Pitfalls

An important goal of school mathematics involves helping students use the powerful forms of representation that have been developed over the centuries through the work of mathematicians throughout the world. However, challenges exist in encouraging students to develop meaning for the mathematical symbols used in formal algebra. Research has demonstrated that students often fail to develop a deep understanding of the meaning of symbolic representations of variables (e.g., Booth 1984; Clement 1982), so much so that Thompson (1994) found that a limited understanding of the meaning of variables negatively impacts students who later take college calculus. The question arises as to how we can develop meaning for formal algebraic symbols in the middle grades so that instruction can build on this meaning throughout students' high school and college experiences.

Developing Algebraic Reasoning Through Generalization

NCTM's (2000) recommendations for algebra in the middle grades strive to assist students' transition to formal algebra by developing meaning for the algebraic symbols that students use. Further, students are expected to have opportunities to develop understanding of patterns and functions, represent and analyze mathematical situations, develop mathematical models, and analyze change. By helping students move from specific numeric situations to develop general rules that model all situations of that type, teachers in fact begin to address the NCTM's recommendations for algebra. Generalizing numeric situations can create strong connections between the mathematical content strands of number and operation and algebra (as well as with other content strands). In addition, these generalizing activities build on what students already know about number and operation and can help students develop a deeper understanding of formal algebraic symbols.

Some Simple Logical Notions Encountered in Elementary Mathematics

It is a serious shortcoming of our elementary texts that they fail to explain the logical notions which underlie the processes employed in proving identities and in solving equations. The situation might be less embarrassing were it not met so early. But it is at the very beginning of the study of formal algebra that it becomes necessary to distinguish between, and to deal with, identical equations and conditional equations; and, following closely upon the course in algebra, comes trigonometry with its chapter on identities.