# Number Sense

## Second and Fifth Graders’ Use of Knowledge-Pieces and Knowledge-Structures When Solving Integer Addition Problems

In this study, we explored second and fifth graders’ noticing of negative signs and incorporation of them into their strategies when solving integer addition problems. Fifty-one out of 102 second graders and 90 out of 102 fifth graders read or used negative signs at least once across the 11 problems. Among second graders, one of their most common strategies was subtracting numbers using their absolute values, which aligned with students’ whole number knowledge-pieces and knowledge-structure.

Author/Presenter

Laura Bofferding

Year
2021
Short Description

In this study, we explored second and fifth graders’ noticing of negative signs and incorporation of them into their strategies when solving integer addition problems. For both grade levels, the order of the numerals, the location of the negative signs, and also the numbers’ absolute values in the problems played a role in students’ strategies used. Fifth graders’ greater strategy variability often reflected strategic use of the meanings of the minus sign. Our findings provide insights into students’ problem interpretation and solution strategies for integer addition problems and supports a blended theory of conceptual change.

## The Development and Assessment of Counting-based Cardinal Number Concepts

The give-n task is widely used in developmental psychology to indicate young children’s knowledge or use of the cardinality principle (CP): the last number word used in the counting process indicates the total number of items in a collection. Fuson (1988) distinguished between the CP, which she called the count-cardinal concept, and the cardinal-count concept, which she argued is a more advanced cardinality concept that underlies the counting-out process required by the give-n task with larger numbers.

Author/Presenter

Arthur J. Baroody

Menglung Lai

Year
2022
Short Description

The give-n task is widely used in developmental psychology to indicate young children’s knowledge or use of the cardinality principle (CP): the last number word used in the counting process indicates the total number of items in a collection. Fuson (1988) distinguished between the CP, which she called the count-cardinal concept, and the cardinal-count concept, which she argued is a more advanced cardinality concept that underlies the counting-out process required by the give-n task with larger numbers. One aim of the present research was to evaluate Fuson’s disputed hypothesis that these two cardinality concepts are distinct and that the count-cardinal concept serves as a developmental prerequisite for constructing the cardinal-count concept. Consistent with Fuson’s hypothesis, the present study with twenty-four 3- and 4-year-olds revealed that success on a battery of tests assessing understanding of the count-cardinal concept was significantly and substantially better than that on the give-n task, which she presumed assessed the cardinal-count concept.

## Teaching Practices for Differentiating Mathematics Instruction for Middle School Students

Three iterative, 18-episode design experiments were conducted after school with groups of 6–9 middle school students to understand how to differentiate mathematics instruction. Prior research on differentiating instruction (DI) and hypothetical learning trajectories guided the instruction. As the experiments proceeded, this definition of DI emerged: proactively tailoring instruction to students’ mathematical thinking while developing a cohesive classroom community.

Author/Presenter

Amy J. Hackenberg

Mark Creager

Ayfer Eker

Year
2020
Short Description

This study is a case of using second-order models of students’ mathematical thinking to differentiate instruction, and it reveals that inquiring into research-based knowledge and inquiring responsively into students’ thinking are at the heart of differentiating mathematics instruction.

## Children’s Integer Understanding and the Effects of Linear Board Games: A Look at Two Measures

The purpose of this study was to identify affordances and limitations of using order and value comparison tasks versus number placement tasks to infer students’ negative integer understanding and growth in understanding. Data came from an experiment with kindergarteners (N = 45) and first graders (N = 48), where the experimental group played a numerical linear board game and the other group did control activities, both involving negative integers.

Author/Presenter

Laura Bofferding

Andrew Hoffman