Jessica Pierson Bishop

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.

Mathematical Discourse as a Process that Mediates Learning in SimCalc Classrooms

While the infrastructure of the SimCalc technology provides unique opportunities for students to wrestle with complex and conceptually difficult mathematics in developmentally appropriate ways, the degree to which students engage in “doing” mathematics varies widely across classrooms. Some classrooms provide minimal opportunities for students to conjecture, justify, and exemplify, whereas in other classrooms these activities are an indispensable part of their day-to-day operations. One of the primary mechanisms by which students can be encouraged to engage in authentic mathematical activities is through the use of appropriately challenging discourse. In this chapter, I describe discursive norms and patterns observed across multiple SimCalc classrooms in order to better understand the interaction between discourse and technology in the broader classroom environment.

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.