The ‘mound-hollow’ model for solving integer addition and subtraction problems

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.