Lisa L. Lamb

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.

Challenges in Measuring Secondary Mathematics Teachers’ Professional Noticing of Students’ Mathematical Thinking

Our focus is on teachers’ professional noticing of children’s mathematical thinking which Jacobs et al. (Jacobs, Lamb, & Philipp, 2010) describe as a set of three interrelated skills: (1) attending to students’ strategies, (2) interpreting the students’ mathematical understandings, (3) deciding how to respond on the basis of students’ understandings. We focus on secondary teachers’ professional noticing of children’s mathematical thinking because we believe teachers with expertise in noticing children’s mathematical thinking are better poised to support their students’ mathematical performance and understanding. However, studying teacher noticing at the secondary level presents unique methodological challenges. We first consider methodological issues in measuring K–12 teachers’ professional noticing of children’s mathematical thinking, and then consider three methodological challenges that are particular to secondary mathematics classrooms. These issues center around the challenges of artifact selection, determining the relative sophistication of responses, and the lack of access to experts’ responses at the secondary level. Also, we consider the cultures of teaching in the elementary and secondary contexts.

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.