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Shutler, Paul Maurice Edmund
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The ‘mound-hollow’ model for solving integer addition and subtraction problems Sari, Puspita; Dindyal, Jaguthsing; Shutler, Paul Maurice Edmund
Journal on Mathematics Education Vol. 16 No. 1 (2025): Journal on Mathematics Education
Publisher : Universitas Sriwijaya in collaboration with Indonesian Mathematical Society (IndoMS)

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.22342/jme.v16i1.pp365-382

Abstract

Understanding integer operations is a fundamental yet challenging concept for elementary students, often requiring effective visual models to support their comprehension. Despite various instructional models, many students continue to struggle with integer addition and subtraction, particularly when negative numbers are involved. Addressing this gap, this study explores the potential of the mound-hollow model to facilitate students’ understanding of integer addition and subtraction concepts intuitively. This study aimed to examine how three sixth-grade students utilized the mound-hollow model to solve integer addition and subtraction problems. Data were collected from students' written tests and individual interviews conducted after a teaching experiment involving 25 sixth graders in Indonesia. The findings indicate that the mound-hollow model provides a meaningful analogy for solving addition problems of types x+(-y) and (-x)+y (where x>y and x,y are natural numbers) and subtraction problems of types x-(-y) and (-x)-y. All three students successfully employed the model to solve the addition problems by neutralizing every mound-hollow pair and to solve the subtraction problems by creating mound-hollow pairs. Additionally, students demonstrated the ability to justify their solutions and correct errors through the mound-hollow representation. The use of a single mound or hollow to represent larger integers enhanced students’ proficiency in solving integer operations and reinforced their understanding of the relationship between addition and subtraction, such as x-(-y)=x+y and (-x)-y=(-x)+(-y). These findings highlight the effectiveness of the mound-hollow model as an alternative instructional tool for teaching integer operations, providing students with an intuitive framework to construct abstract mathematical concepts. The implications of this study contribute to mathematics education by offering insights into the design of visual models that support conceptual understanding in integer arithmetic.
Types of Students’ Reasoning and Difficulties in Solving Integer Addition and Subtraction Problems Sari, Puspita; Shutler, Paul Maurice Edmund
Mathematics Education Journal Vol. 20 No. 1 (2026): Mathematics Education Journal
Publisher : Universitas Sriwijaya in collaboration with Indonesian Mathematical Society (IndoMS)

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.22342/mej.v20i1.pp51-70

Abstract

Students' well-documented difficulties and errors with integer arithmetic often stem from a reliance on procedural rules over conceptual understanding. Although numerous studies have examined various types of students’ reasoning and difficulties involved in solving integer addition and subtraction problems, existing research remains fragmented, and there is a lack of a systematic synthesis that explicitly links types of students’ reasoning to the specific difficulties they experience across different contexts and grade levels. This systematic literature review (SLR) paper synthesizes findings from 45 studies to describe types of students’ reasoning and the associated difficulties in solving integer addition and subtraction problems. Data was derived from both exploratory and experimental studies that provide descriptions on how students in grade 1 to 8 reason when solving integer addition and subtraction problems.  Data collection method involved identifying peer-reviewed journal articles through database searches and reference tracing, guided by specific inclusion and exclusion criteria. Across three broad categories of students’ reasoning-magnitude-based, order-based, and symbolic-logical-this review offers a synthesized framework that characterizes students’ conceptions of negative integers, strategies in solving integer addition and subtraction problems, and difficulties students encounter in the reasoning process. The findings indicate that students’ difficulties stem from their conceptions of integers within specific reasoning types, which often conflict with their prior understanding of whole numbers. This literature review also found that students sometimes apply reasoning that does not align with the context of a given problem. Such misalignment reflects students’ misconceptions, which contribute to difficulties in solving problems accurately and meaningfully.
Co-Authors Dindyal, Jaguthsing Puspita Sari