Search

The fundamental aim in this article is to elucidate cognitive factors that influence the development of mathematical structures and incipient generalizations in elementary school children on the basis of their work on patterns, including how they use various representational forms such as gestures, words, and arithmetical symbols to convey their expressions of generality. We describe approximate and exact pattern generalizations and three cognitive factors that mutually influence the emergence of mathematical structures, namely, competence with number relationships, competence with shape similarity, and competence with figural property construction, discernment, and justification. We also highlight various representational modes that elementary students use to capture their emergent structures and incipient generalizations, grade-appropriate use and understanding of variables via the notions of intuited and tacit variables, and ways in which their structural incipient generalizations support their early understanding of functions.

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the domains of functions and models in the CCSSM from Grade 5 to Grade 8 and Algebra 1. Table 9.1 lists the appropriate pages for your convenience. Functions in the CCSSM are formally defined in eighth grade – that is, as rules that assign to each input exactly one output.

Like physical and concrete manipulatives, technology-mediated tools for teaching and learning middle school mathematical concepts and processes provide students with an opportunity to actively engage in thinking while they tinker with the relevant objects in a virtual context.

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the domains of data, statistics, and probability in the CCSSM from Grade 5 to Grade 8 and Algebra 1. Table 11.1 lists the appropriate pages for your convenience. Since data, statistics, and probability involve relationships between quantities, middle school students can use descriptive and analytic modeling processes to represent and analyze those relationships in a structured way.

This methods book takes a very practical approach to learning to teach middle school school mathematics in the Age of the Common Core State Standards (CCSS). The Kindergarten through Grade 12 CCSS in Mathematics (i.e., CCSSM) was officially released on June 2, 2010 with 45 of the 50 US states in agreement to adopt it. Consequently, that action also meant implementing changes in their respective state standards and curriculum in mathematics.

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the Ratio and Proportion (RP) and Quantities (N-Q).domains of the CCSSM from Grades 6 to 8 and Algebra 1.

In this chapter you will deal with content-practice, teaching, and learning issues relevant to constructing and manipulating expressions from Grade 5 to Grade 8 and Algebra 1 of the CCSSM. Table 7.1 lists the specific domains and relevant content standards that are covered in this chapter.

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the Number System (NS) domain of the CCSSM from Grade 6 to Grade 8 and Algebra 1. In particular, this chapter will help you establish connections between whole numbers and fractions in the earlier grades and integers, rational numbers, and irrational numbers in the middle grades.

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the domains of geometry and measurement from Grade 5 to Grade 8 of the CCSSM. Table 10.1 lists the appropriate pages for your convenience. The development of fifth-grade students’ mathematical understanding of measurement follows the same multiplicative structure that they already learned in the elementary grades (i.e., identify a (standard) unit and perform repeated iterations of the same unit).

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the following CCSSM number-related domains: Numbers and Operations in Base Ten (NBT) in Grade 5; Numbers and Operations – Fractions (NF) in Grade 5; and portions of Number System (NS) in Grade 6. In Chapter 4, you will reconceptualize them in terms of their relationships with other structures of numbers (i.e., integers, rational numbers, irrational numbers, and real numbers).

In this chapter, you will deal with content-practice, teaching, and learning issues relevant to the four arithmetical operations of addition, subtraction, multiplication, and division involving whole numbers greater than 10 and decimal numbers up to the hundredths place.

In this chapter, you will deal with content-practice, teaching, and learning issues relevant to fractions and operations in the Numbers and Operations –Fractions (NF) domain from Grades 1 through 5 of the CCSSM.

In this chapter you will deal with various issues relevant to setting up and running an effective content-practice driven mathematics classroom. Such classrooms typically provide a climate that is conducive for teaching and learning mathematics, that is, a sociocultural condition that supports meaningful engagement with the content and practice standards.

In this chapter, you will deal with content-practice, teaching, and learning issues relevant to the Measurement and Data (MD) domain from kindergarten (K) through grade 5 of the CCSSM. The measurement clusters build on multiplicative structures, while the data clusters focus on ways in which data can be organized and reasonably interpreted around well-defined structures.