Article Sidebar

Download
Statistic
Read Counter : 30

Downloads

Download data is not yet available.

Main Article Content

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 approaches, 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 as an intuitive representation to facilitate students’ understanding of integer operations. 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 for addition by neutralizing every mound-hollow pair and used diagrammatic representations to solve subtraction problems by forming corresponding 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.

Keywords

Addition Integers Mound-Hollow Model Negative Subtraction

Article Details

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite
Sari, P., Dindyal, J., & Shutler, P. M. E. (2025). The ‘mound-hollow’ model for solving integer addition and subtraction problems. Journal on Mathematics Education, 16(1), 365–382. https://doi.org/10.22342/jme.v16i1.pp365-382
Download Citation
Endnote/Zotero/Mendeley (RIS)
BibTeX

References

  1. Aqazade, M., & Bofferding, L. (2021). Second and Fifth Graders’ Use of Knowledge-Pieces and Knowledge-Structures When Solving Integer Addition Problems. Journal of Numerical Cognition, 7(2), 82-103. https://doi.org/10.5964/jnc.6563
  2. Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93(4), 373-397. https://doi.org/10.1086/461730
  3. Battista, M. T. (1983). A complete model for operations on integers. Arithmetic Teacher, 30(9), 26-31. https://doi.org/10.5951/AT.30.9.0026
  4. Bishop, J. P., Lamb, L. L., Philipp, R. A., Whitacre, I., Schappelle, B. P., & Lewis, M. L. (2014). Obstacles and affordances for integer reasoning: An analysis of children's thinking and the history of mathematics. Journal for Research in Mathematics Education, 45(1), 19-61. https://doi.org/10.5951/jresematheduc.45.1.0019
  5. Bofferding, L. (2014). Negative integer understanding: Characterizing first graders’ mental models. Journal for Research in Mathematics Education, 45(2), 194-245. https://doi.org/10.5951/jresematheduc.45.2.0194
  6. Bofferding (2019). Understanding negative numbers. In A. Norton & M. W. Alibali (Eds.), Constructing number: Merging perspectives from psychology and mathematics education (pp. 251-277). Springer International Publishing AG.
  7. Bofferding, L., Aqazade, M., & Farmer, S. (2017). Second graders' integer addition understanding: Leveraging contrasting cases. In E. Galindo & J. Newton (Eds.), Proceedings of the 39th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. (pp. 243-250). Hoosier Association of Mathematics Teacher Educators.
  8. Bolt, A. B., Holding, D. J., Tammadge, A. R., & Tyson, J. V. (1965). The school mathematics project book 1 (A. G. Howson, Ed.). Cambridge University Press
  9. Bolyard, J., & Moyer-Packenham, P. S. (2012). Making sense of integer arithmetic: The effect of using virtual manipulatives on students’ representational fluency. Journal of Computers in Mathematics and Science Teaching, 31(2), 93-113.
  10. Clements, D. H., & McMillen, S. (1996). Rethinking 'concrete' manipulatives. Teaching Children Mathematics, 2(5), 270-279.
  11. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Reidel.
  12. Gallardo, A., & Rojano, T. (1994). School algebra syntactic difficulties in the operativity with negative numbers. In D. Kirshner (Ed.), The Sixteenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 159-165). Louisiana State University.
  13. Goldin, G.A., & Shteingold, N. (2001). Systems of representations and the development of mathematical concepts. In A. A. Cuoco & F. R. Cuoco (Eds.), The roles of representation in school mathematics (2001 yearbook) (pp. 1-23). The National Council of Teachers of Mathematics, Inc.
  14. Haylock, D., & Cockburn, A. (1997). Understanding mathematics in the lower primary years (Revised and expanded ed.). Paul Chapman Publishing Ltd.
  15. Janvier, C. (1985). Comparison of models aimed at teaching signed integers. In L. Streefland (Ed.), Proceedings of the Ninth Annual Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 135-140). International Group for the Psychology of Mathematics Education.
  16. Kamii, C., Lewis, B. A., & Kirkland, L. (2001). Manipulatives: When are they useful? Journal of Mathematical Behavior, 20, 21-31
  17. Kilhamn, C. (2018). Different differences: Metaphorical interpretations of “difference” in integer addition and subtraction. In L. Bofferding & N. Wessman-Enzinger (Eds.), Exploring the integer addition and subtraction landscape (pp. 143-166). Springer.
  18. Kilpatrick, J., Swafford, J., & Bradford Findell. (2001). Adding it up : Helping children learn mathematics. National Academy Press.
  19. Küchemann, D. (1981). Positive and negative numbers. In K. M. Hart (Ed.), Children’s understanding of mathematics (Vol. 11-16, pp. 82-87). CSMS Mathematics Team.
  20. Lesh, R., & Doerr, H. M. (2000). Symbolizing, communicating, and mathematizing: Key components of models and modeling. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classroom: perspectives on discourse, tools, and instructional design (pp. 361-384). Lawrence Erlbaum Associates.
  21. Liebeck, P. (1990). Scores and forfeits: An intuitive model for integer arithmetic. Educational Studies in Mathematics, 21, 221-239. https://doi.org/10.1007/BF00305091.
  22. Linchevski, L., & Williams, J. (1999). Using intuition from everyday life in 'filling' the gap in children's extension of their number concept to include the negative numbers. Educational Studies in Mathematics, 39, 131-147.
  23. Lytle, P. A. (1994). Investigation of a model based on the neutralization of opposites to teach integer addition and subtraction. In J. P. d. Ponte & J. F. Matos (Eds.), Proceedings of the 18th International Conference for the Psychology of Mathematics Education (Vol. 3, pp. 192-199)
  24. Mathison, S. (2005). Encyclopedia of evaluation. Sage Publications, Inc. https://doi.org/10.4135/9781412950558
  25. Merenluoto, K., & Lehtinen, E. (2004). Number concept and conceptual change: Towards a systemic model of the processes of change. Learning and Instruction, 14(5), 519-534. https://doi.org/10.1016/j.learninstruc.2004.06.016
  26. NCTM (The National Council of Teachers of Mathematics). (1970). Positive and negative numbers. In Experiences in mathematical discovery.
  27. New Zealand Government. (2014). Hills and dales. https://nzmaths.co.nz/resource/hills-and-dales
  28. Nurnberger-Haag, J. (2018). Take it away or walk the other way? Finding positive solutions for integer subtraction. In L. Bofferding & N. M. Wessman-Enzinger (Eds.), Exploring the integer addition and subtraction landscape: Perspectives on integer thinking (pp. 109-142). Springer.
  29. Sari, P., Hajizah, M. N., & Purwanto, S. (2020). The neutralization on an empty number line model for integer additions and subtractions: Is it helpful? . Journal on Mathematics Education, 11, 1-16
  30. Sari, P., (2023). Developing students’ understanding of integer addition and subtraction [Doctoral dissertation, National Institue of Education, Nanyang Technological University]
  31. Schwarz, B. B., Kohn, A. S., & Resnick, L. B. (1993-1994). Positives about negatives: A case study of an intermediate model for signed numbers. The Journal of the Learning Sciences, 3(1), 37-92.
  32. Shutler, P. M. E. (2017). A symbolical approach to negative numbers. The Mathematics Enthusiast, 14(nos1,2&3), 207-240.
  33. Stephan, M., & Akyuz, D. (2012). A proposed instructional theory for integer addition and subtraction. Journal for Research in Mathematics Education 43(4), 428-464. https://doi.org/10.5951/jresematheduc.43.4.0428
  34. Thompson, P. W., & Dreyfus, T. (1988). Integers as transformations. Journal for Research in Mathematics Education 19(2), 115-133. https://doi.org/10.5951/jresematheduc.19.2.0115
  35. Ulrich, C. (2012). The addition and subtraction of signed quantities. In L. H. R. Mayes, & M. Mackritis (Ed.), Quantitative reasoning and mathematical modeling: A driver for STEM integrated education and teaching in context (pp. 127-141). University of Wyoming.
  36. Vig, R., Murray, E. & Star, J.R. (2014). Model breaking points conceptualized. Educational Psychology Review 26, 73–90. https://doi.org/10.1007/s10648-014-9254-6
  37. Vlassis, J. l. (2004). Making sense of the minus sign or becoming flexible in ‘negativity’. Learning and Instruction, 14, 469-484. https://doi.org/10.1016/j.learninstruc.2004.06.012
  38. Vlassis, J. l. (2008). The role of mathematical symbols in the development of number conceptualization: The case of the minus sign. Philosophical Psychology, 21(4), 555-570. https://doi.org/10.1080/09515080802285552
  39. Vosniadou, S., & Verschaffel, L. (2004). Extending the conceptual change approach to mathematics learning and teaching [Editorial]. Learning and Instruction, 14(5), 445-451. https://doi.org/https://doi.org/10.1016/j.learninstruc.2004.06.014
  40. Wessman-Enzinger, N. M. & Mooney, E. S. (2014). Making sense of integers through storytelling. Mathematics Teaching in the Middle School, 20 (4), 202-205 https://doi.org/10.5951/mathteacmiddscho.20.4.0202
  41. Wessman-Enzinger, N. M., & Mooney, E. S. (2019). Conceptual models for integer addition and subtraction. International Journal of Mathematical Education in Science and Technology, 52(3), 349–376. https://doi.org/10.1080/0020739X.2019.1685136
  42. Whitacre, I., Bishop, J. P., Lamb, L. L. C., Philipp, R. A., Schappelle, B. P., & Lewis, M. L. (2012). Happy and sad thoughts: An exploration of children’s integer reasoning. Journal of Mathematical Behavior, 31, 356-365