Bonnie P. Schappelle

Setting the Standards

On March 21, 1989, the National Council of Teachers of Mathematics (NCTM) held a press conference in Washington, DC, to release the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989). The Curriculum and Evaluation Standards had been under development for almost five years (Crosswhite, Dossey, & Frye, 1989), and public interest was high; the press conference, which some had expected to be quite small, was moved to the larger venue of the Willard Hotel in downtown Washington. Since NCTM had not organized such a large media event before, it was an exciting time for the organization. As Marilyn Hala, an NCTM staff member, recalled: This is the first time I ever remember that we had a bank of six or seven video cameras in the back of the room. There were probably 200 people at the press conference in the Willard Ballroom. It was quite impressive! You had to walk by these members of the press — these cameras — to sit down.1

Conceptions and practices of extraordinary mathematics teachers

Abstract Four teachers were identified as “extraordinary“ teachers of mathematics. Data gathered from interviews, tests on content knowledge, discussions during a series of seminars, and one observation in each teachers classroom were used in summarizing characteristics of these teachers under three headings: (a) their mathematical preparation and their content knowledge of mathematics: (b) their conceptions about mathematics, about learning, about teaching, about the roles of teachers and of students, and about the assessment of learning; and (c) their teaching practices. Within each of these three areas, many commonalities were found among these teachers. This article will describe these commonalities and make inferences about the conditions and support necessary for teachers to begin to adopt beliefs and practices necessary to sustain reform of school mathematics.

Leveraging Structure: Logical Necessity in the Context of Integer Arithmetic

ABSTRACT Looking for, recognizing, and using underlying mathematical structure is an important aspect of mathematical reasoning. We explore the use of mathematical structure in children’s integer strategies by developing and exemplifying the construct of logical necessity. Students in our study used logical necessity to approach and use numbers in a formal, algebraic way, leveraging key mathematical ideas about inverses, the structure of our number system, and fundamental properties. We identified the use of carefully chosen comparisons as a key feature of logical necessity and documented three types of comparisons students made when solving integer tasks. We believe that logical necessity can be applied in various mathematical domains to support students to successfully engage with mathematical structure across the K–12 curriculum.

‘Negative of my money, positive of her money’: secondary students’ ways of relating equations to a debt context

We interviewed 40 students each in grades 7 and 11 to investigate their integer-related reasoning. In one task, the students were asked to write and interpret equations related to a story problem about borrowing money from a friend. All the students solved the story problem correctly. However, they reasoned about the problem in different ways. Many students represented the situation numerically without invoking negative numbers, whereas others wrote equations involving negative numbers. When asked to interpret equations involving negative numbers in relation to the story, students did so in two ways. Their responses reflect distinct perspectives concerning the relationship between arithmetic equations and borrowing/owing. We discuss these findings and their implications regarding the role of contexts in integer instruction.

Students’ Thinking About Integer Open Number Sentences

We share a subset of the 41 underlying strategies that comprise five ways of reasoning about integer addition and subtraction: formal, order-based, analogy-based, computational, and emergent. The examples of the strategies are designed to provide clear comparisons and contrasts to support both teachers and researchers in understanding specific strategies within the ways of reasoning. The ability to categorize strategies into one of five ways of reasoning may enable teachers to organize knowledge of student thinking in ways that are useable and accessible for them and provide researchers with sufficient information about the strategies and ways of reasoning such that they can reliably build on this work.