Understanding Error Patterns in Integer Operations of Students With and Without Mathematics Difficulty: A Descriptive Analysis

Abstract

This study examined error patterns in integer addition and subtraction among 160 seventh-grade students, including students with and without mathematics difficulty (MD). A quantitative analysis identified common errors; in addition, qualitative interviews with three students with MD provided deeper insights. All students struggled more with integer subtraction than addition, particularly (−2) − (−3). Students with MD made more errors than their peers without MD on both integer addition and subtraction. While both groups frequently made conceptual and procedural errors in subtraction, students with MD exhibited more severe misconceptions in addition. Key error patterns included misinterpreting signs, determining negativity based on parentheses, neglecting unary/binary signs, omitting negative signs in answers, relying on rote memorization, and misusing the number line. These findings highlighting persistent misconceptions in integer operations, particularly among students with MD, emphasize the need for targeted instructional support to address these challenges effectively.

Keywords

Integer operations math difficulty error patterns

Get full access to this article

View all access options for this article.

References

Aqazade

Bofferding

Farmer

(2017). Learning integer addition: is later better? North American Chapter of the International Group for the Psychology of Mathematics Education.

Ashlock

R. B.

(2010). Error Patterns in Computation: Using Error Patterns to Help Each Student Learn (10th ed.). Allyn & Bacon.

Barbieri

C. A.

Rodrigues

(2025). Leveraging cognitive load theory to support students with mathematics difficulty. Educational Psychologist, 60(3), 208–32. https://doi.org/10.1080/00461520.2025.2486138

Bishop

J. P.

Lamb

L. L.

Philipp

R. A.

Whitacre

Schappelle

B. P.

(2018). Students’ thinking about integer open number sentences. In Bofferding

Wessman Enzinger

N. M.

(Eds.), Exploring the integer Addition and Subtraction Landscape: Perspectives on integer Thinking (pp. 47–71). Springer. https://doi.org/10.1007/978-3-319- 90692-8_3.

Bishop

J. P.

Lamb

L. L.

Philipp

R. A.

Whitacre

Schappelle

B. P.

Lewis

M. L.

(2014). Obstacles and affordances for integer reasoning: an analysis of children’s thinking and the history of mathematics. J Res Math Educ, 45(1), 19–61. https://doi.org/10.5951/jresematheduc.45.1.0019

Bofferding

(2014). Negative integer understanding: characterizing first graders’ mental models. J Res Math Educ, 45(2), 194–245. https://doi.org/10.5951/jresematheduc.45.2.0194

Bofferding

(2018). Challenges of promoting conceptual change with instructional contexts. In Bofferding

Wessman-Enzinger

N. M.

(Eds.), Exploring the integer Addition and Subtraction Landscape: Perspectives on integer Thinking (pp. 167–87). Springer Nature. https://doi.org/10.1007/978-3-319-90692-8_7

Bofferding

Richardson

S. E.

(2013). Investigating integer addition and subtraction: a task analysis. North American Chapter of the International Group for the Psychology of Mathematics Education.

Bofferding

Wessman-Enzinger

2018). Exploring the integer addition and subtraction landscape perspectives on integer thinking. Retrieved from https://doi.org/10.1007/978-3-319-90692-8

10.

Bouck

E. C.

Long

Park

(2021). Using a virtual number line and corrective feedback to teach addition of integers to middle school students with developmental disabilities. J Dev Phys Disabil, 33(1), 99–116. https://doi.org/10.1007/s10882-020-09735-z

11.

Butterworth

(2010). Foundational numerical capacities and the origins of dyscalculia. Trends Cogn Sci, 14(12), 534–41. https://doi.org/10.1016/j.tics.2010.09.007

12.

Carlson

J. A.

(2010). Avoiding traps in member checking. Qualitative Report, 15(5), 1102–13. https://doi.org/10.46743/2160-3715/2010.1332

13.

Cirino

P. T.

Tolar

T. D.

Fuchs

L. S.

Huston-Warren

(2016). Cognitive and numerosity predictors of mathematical skills in middle school. J Exp Child Psychol, 145(4), 95–119. https://doi.org/10.1016/j.jecp.2015.12.010

14.

Geary

D. C.

(2004). Mathematics and learning disabilities. J Learn Disabil, 37(1), 4–15. https://doi.org/10.1177/00222194040370010201

15.

Gesuelli

Miller-Cotto

Barbieri

C. A.

(2025). Variability in math achievement growth among students with early math learning difficulties and the role of school supports. J Educ Psychol. https://doi.org/10.1037/edu0000928

16.

Hawthorne

Philipp

R. A.

Lamb

L. L.

Bishop

J. P.

Whitacre

Schappelle

B. P.

(2022). Reconceptualizing a mathematical domain on the basis of student reasoning: considering teachers’ perspectives about integers. The Journal of Mathematical Behavior, 65(6), 100931. https://doi.org/10.1016/j.jmathb.2021.100931

17.

Hughes

E. M.

Witzel

B. S.

Myers

Lin

T. H.

(2023). Unpacking and understanding specific learning disabilities in mathematics. Teach Except Child, 56(1), 26–32. https://doi.org/10.1177/00400599231175509

18.

Hwang

Riccomini

P. J.

(2021). A descriptive analysis of the error patterns observed in the fraction-computation solution pathways of students with and without learning disabilities. Assess Eff Interv, 46(2), 132–42. https://doi.org/10.1177/1534508419872256

19.

Individuals with Disabilities Education Act (IDEA) . (2004). 20 U.S.C., § 1400.

20.

Khalid

Embong

(2020). Sources and possible causes of errors and misconceptions in operations of integers. International Electronic Journal of Mathematics Education, 15(2), em0568. https://doi.org/10.29333/iejme/6265

21.

Kingsdorf

Krawec

(2014). Error analysis of mathematical word problem solving across students with and without learning disabilities. Learning Disabilities Research and Practice, 29(2), 66–74. https://doi.org/10.1111/ldrp.12029

22.

Lamb

Bishop

Whitacre

Philipp

(2023). Flexibility across and flexibility within: the domain of integer addition and subtraction. The Journal of Mathematical Behavior, 70(2), 101031. https://doi.org/10.1016/j.jmathb.2023.101031

23.

Lewis

K. E.

Fisher

M. B.

(2018). Clinical interviews: assessing and designing mathematics instruction for students with disabilities. Interv Sch Clin, 53(5), 283–91. https://doi.org/10.1177/1053451217736864

24.

Lewis

K. E.

Lynn

D. M.

(2018). Against the odds: insights from a statistician with dyscalculia. Education Sciences, 8(2), 63. https://doi.org/10.3390/educsci8020063

25.

Lewis

K. E.

Sweeney

Thompson

G. M.

Adler

R. M.

(2020). Integer number sense and notation: a case study of a student with a mathematics learning disability. The Journal of Mathematical Behavior, 59(4), 100797. https://doi.org/10.1016/j.jmathb.2020.100797

26.

Makonye

J. P.

Fakude

(2016). A study of errors and misconceptions in the learning of addition and subtraction of directed numbers in grade 8. SAGE Open, 6(4), 2158244016671375. https://doi.org/10.1177/2158244016671375

27.

Mussolin

De Volder

Grandin

Schlogel

Nassogne

M. C.

Noël

M. P.

(2010). Neural correlates of symbolic number comparison in developmental dyscalculia. J Cogn Neurosci, 22(5), 860–74. https://doi.org/https://https://doi.org/10.1162/https://doi.org/ jocn.2009.21237

28.

National Governors Association Center for Best Practices & Council of Chief State School Officers . (2010). Common core state standards for mathematics.

29.

Nelson

Powell

S. R.

(2018). Computation error analysis: students with mathematics difficulty compared to typically achieving students. Assess Eff Interv, 43(3), 144–56. https://doi.org/10.1177/1534508417745627

30.

Nowell

L. S.

Norris

J. M.

White

D. E.

Moules

N. J.

(2017). Thematic analysis: striving to meet the trustworthiness criteria. Int J Qual Methods, 16(1), 1–13. https://doi.org/10.1177/1609406917733847

31.

Permata

Wijayanti

, & Masriyah (2019). Students’ misconceptions on the algebraic prerequisites concept: operation of integer numbers and fractions. J Phys: Conf Ser, 1188(1), 12059. https://doi.org/10.1088/1742-6596/1188/1/012059

32.

Piazza

Facoetti

Trussardi

A. N.

Berteletti

Conte

Lucangeli

Dehaene

Zorzi

(2010). Developmental trajectory of number acuity reveals a severe impairment in developmental dyscalculia. Cognition, 116(1), 33–41. https://doi.org/10.1016/j.cognition.2010.03.012

33.

Rosyidah

A. N. K.

Maulyda

M. A.

Jiwandono

I. S.

Oktaviyanti

Gunawan

(2021). Misconceptions and errors in integer operations: a study in preservice elementary school teachers (PGSD). Journal of Physics. Conference Series, 1779(1), 12078. https://doi.org/10.1088/1742-6596/1779/1/012078

34.

Rousselle

Noël

M. P.

(2007). Basic numerical skills in children with mathematics learning disabilities: a com parison of symbolic vs non-symbolic number magnitude processing. Cognition, 102(3), 361–95. https://doi.org/10.1016/j.cognition.2006.01.005

35.

Sharpe

S. T.

Krawec

Fults

(2014). The effects of a mathematical problem-solving intervention on the errors made by middle school students with and without learning disabilities. Middle Grades Research Journal, 9(3), 43.

36.

Stephan

Akyuz

(2018). Teaching integers to students with disabilities: three case studies. In Bofferding

Wessman-Enzinger

N. M.

(Eds.), Exploring the integer Addition and Subtraction Landscape (pp. 75–108). Springer International Publishing. https://doi.org/10.1007/978-3-319-90692-8_4

37.

Swanson

H. L.

Howard

C. B.

Saez

(2006). Do different components of working memory underlie different subgroups of Reading disabilities? J Learn Disabil, 39(3), 252–69. https://doi.org/10.1177/00222194060390030501

38.

Sweller

(2006). The worked example effect and human cognition. Learn Instr, 16(2), 165–9. https://doi.org/10.1016/j.learninstruc.2006.02.005

39.

Utomo

D. P.

(2020). The pattern of a relational understanding of fifth-grade students on integer operations. Journal of Research and Advances in Mathematics Education, 5(2), 119–29. https://doi.org/10.23917/jramathedu.v5i2.954

40.

Van Luit

J. E.

(2019). Diagnostics of dyscalculia. In International Hand Book of Mathematical Learning Difficulties: From the Laboratory to the Classroom (pp. 653–68). Springer.

41.

Wessman-Enzinger

N. M.

(2019). Grade 5 children’s drawings for integer addition and subtraction open number sentences. The Journal of Mathematical Behavior, 53(9), 105–28. https://doi.org/10.1016/j.jmathb.2018.03.010

42.

Whitacre

Bishop

J. P.

Philipp

R. A.

Lamb

L. L.

Schappelle

B. P.

(2014). Dollars & sense: students’ integer perspectives. Mathematics Teaching in the Middle School, 20(2), 84–9. https://doi.org/10.5951/mathteacmiddscho.20.2.0084

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

0.09 MB