Subtraction involving negative numbers: Connecting to whole number reasoning

In this article, we explore how students attempt to bridge from their whole number reasoning to integer reasoning as they solve subtraction problems involving negative numbers. Based on interviews with students ranging from first graders to preservice teachers, we identify two overarching strategies: making connections to known problem types and leveraging conceptions of subtraction. Their initial connections suggest that rather than identifying the best instructional models to teach integer concepts, we should focus on identifying integer instructional models that build on the potentially productive connections that students’ already make; we propose an example of one such form of instruction

Examining Pinterest as a Curriculum Resource for Negative Integers: An Initial Investigation

This paper reports an investigation of mathematical resources available on the social media site Pinterest. Pinterest is an online bulletin board where users create visual bookmarks called pins in order to share digital content (e.g., webpages, images, videos). Although recent surveys have shown that Pinterest is a popular reference for teachers, understanding of the mathematical resources available on the site is lacking. To take initial steps in investigating the curriculum resources provided by Pinterest, we used keyword searches to gather a database of pins related to the topic of negative integers. A content analysis was conducted on the pins with a focus on several characteristics including mathematical operations, mathematical models, use of real-world context, and whether mathematical errors were present in source material. Results show a dominance of addition and subtraction over other operations, use of mathematical models in half of pins, infrequent use of real-world context, and mathematical errors in roughly one-third of pins. We provide a breakdown of these results and discuss implications of the findings for mathematics teacher education and professional development

Alice\u27s Drawings for Integer Addition and Subtraction Open Number Sentences

Alice, a fifth grader who participated in twelve weeks of a teaching experiment on integer addition and subtraction, produced drawings as part of her strategy for solving integer addition and subtraction open number sentences. The drawings she created during the twelve weeks of the teaching experiment were analyzed and grouped into the following categories: Single Set of Objects, Double Set of Objects, Number Paths & Number Lines, and Number Sentences. These drawings provide insight into how children may directly model or count when solving integer addition and subtraction problems

Grade 5 Children’s Drawings for Integer Addition and Subtraction Open Number Sentences

Three Grade 5 children participated in a microgenetic study embedded in 12-week teaching experiment on integer addition and subtraction. They solved open number sentences in four individual sessions across the 12-weeks and produced drawings. Through the lens of learner-generated drawings and qualitative analysis, these drawings provide perspective into the children’s thinking about integer addition and subtraction. The following categories are described: Single and Double Set of Objects, Number Sequences, Empty Number Lines, Number Lines, Number Sentences, Sign Emphasis, and Answer in Box Only. One student drew sets of objects frequently and the other students drew number lines more. Descriptions of how use of their drawings changed over time are provided. Implications point to a re-examination of integer instructional models and insight into potential learning progressions

Integers as Directed Quantities (Chapter 13 in Constructing Number)

Mathematics education researchers have long pursued—and many still pursue—an ideal instructional model for operations on integers. In this chapter, I argue that such a pursuit may be futile. Additionally, I highlight that ideas of relativity have been overlooked; and, I contend that current uses of translation within current integer instructional models do not align with learners’ inventions. Yet, conceptions of relativity and translation are essential for making sense of integers as directed quantities. I advocate for drawing on learners’ unique conceptions and actions about directed number in developing instructional models. Providing evidence of student work from my research, I illustrate the powerful constructions of relativity and translation as students engage with directed quantities

An Investigation of Subtraction Algorithms from the 18th and 19th Centuries

How do you subtract? What is your algorithm of choice? The most common subtraction algorithm in the United States today is the decomposition algorithm. While other algorithms exist and are possibly utilized, no other algorithm dominates the contemporary curriculum, classrooms, and pencils of children in the U.S. more so than the decomposition algorithm. Subtraction is one of the fundamental building blocks of arithmetic. Looking to the past and exploring the history of subtraction algorithms can help us glean knowledge of the intended and possibly implemented curricula in past eras. In this paper, special attention will be given to 18th and 19th century America. The aims of this paper are to identify the different algorithms used during this time period and to discuss implications for the modern teacher

Leveraging Different Perspectives to Explore Student Thinking about Integer Addition and Subtraction

This is the third meeting of a working group on student thinking about integers. The main goal of this working group includes utilizing different theoretical perspectives and methodologies in small groups to design complementary studies, where student thinking about integer addition and subtraction will be explored. This working group aims to provide a space for participants to capitalize on their differences in theoretical perspectives and methodologies to promote productive scholarly discussion about the same research topic, student thinking about integer addition and subtraction. Participants will actively engage in work that progresses towards these studies, with the intent to develop a monograph that highlights this research