# Table 1 Overview of articles in which the balance model (BM) was used for teaching linear equations

Article Rationaleb Appearance Use Students involved Research designc Intervention in comparison group (CG)? Learning outcomes (on linear equation solving unless otherwise specified)
Duration intervention Instructional setting Type of equations
Alibali (1999) PE Drawn 40-min session Individual instruction by teacher 3 + 4 + 5 = __ + 5 Grades 3–5; 143 students Pre-posttest; BMa-group and two comparison groups CG1: received feedback CG2: explanation solution steps • 36% of the BM-group improved
• BM-group outperformed CG1
• CG2 outperformed BM-group
Andrews (2003)   Physical, drawn 1 lesson Classroom instruction by teacher 2x + 5 = x + 8 Grade 7; 4 students
Andrews and Sayers (2012) EQ, LI Drawn 5 lessons Classroom instruction by teacher x + 7 = 9 x − 2 = 10 Grade 8, 3 classes
Araya et al. (2010) PE, MR Drawn 2-h session Classroom instruction by learning movie 2x + 1 = 5 + x Grade 7; 236 students; no previous algebra instruction Posttest; BM-group and comparison group CG: symbolic instruction • Below average to high achieving students of the BM-group outperformed the CG
Austin and Vollrath (1989) PE, LI Physical    3w + 5 = 11 “Introductory algebra students”
Berks and Vlasnik (2014) MR Drawn 1 lesson Classroom instruction by teacher 4x + 2y = 12 y = 2x + 2 Students with some algebra experience
Boulton-Lewis et al. (1997)   Drawn 5 lessons Classroom instruction by teacher 2x + 5 = 17 Grade 8; 21 students Pre-posttest; BM-group   • Few students could model or solve equations with the BM
• Most students could solve equations without the BM
• Students preferred not to use the BM
Brodie and Shalem (2011) EQ Drawn 3–5 lessons Classroom instruction by teacher 3 + x = 5 Grade 8
Caglayan and Olive (2010) MR, LI Drawn 2 lessons Classroom instruction by teacher 2c + 1 = 7 2x − 1 = 13 Grade 8; 24 students Descriptive; BM-group   • BM gives meaning to equations with addition/multiplication
• BM does not give meaning to equations with negative values/subtraction
Cooper and Warren (2008) MR Physical; drawn 5 years Classroom instruction by teacher ? + 11 = 36 ?  − 7 = 6 Grades 2–6; 220–270 students Descriptive; BM-group   • Young students can generalize the balance methodd for simple equations
• Older students can generalize the balance method for all operations and use it to solve equations
Figueira-Sampaio, Santos, and Carrijo (2009) EQ, PE, LI Group 1: Physical Group 2: Virtual 50-min lesson Group 1: Classroom instruction by teacher Group 2: Working in pairs with computer 5x + 50 = 3x + 290 Grade 6; 46 students Descriptive; two BM-groups   • Virtual BM-group shows more participation, social interaction, motivation, cooperation, discussion, reflection, and a feeling of responsibility, than the physical BM-group
Filloy and Rojano (1989) EQ, MR, LI Drawn 1 session with 5 problems Individual instruction by teacher 3 + 2x = 5x 10x − 18 = 4x Grade 7; three classes Descriptive; BM-group   • With BM, the step from solving equations with unknowns on one side of the equal sign towards solving equations with unknowns on both sides of the equal sign, is smaller than with the geometrical model
• The geometrical model is more appropriate than the BM for modeling equations with subtraction
• Assigning values to unknowns can hinder students when using the BM
Fyfe, McNeil, and Borjas (2015) PE Physical, drawn 1 lesson Individual instruction by teacher 2 + 3 = 2 + __ Grades 1–3; 389 students
Gavin and Sheffield (2015) EQ Drawn   Classroom instruction by teacher 12 + 23 = 13 + n 51 − n = 50–25 Grade 6; 305 students
Jupri, Drijvers, and Van den Heuvel-Panhuizen (2014)   Drawn 1 item on a test   1 kg + 0.5brick = 1brick Grade 8; 51 students
Kaplan and Alon (2013) PE Virtual 1 session Individual instruction by teacher and individually working with computer ▲▲ =  Grades 3–4; 2 students
Leavy, Hourigan, and McMahon (2013) EQ Physical   Classroom instruction by teacher 8 = __ + 3 Grade 3
Linchevski and Herscovics (1996) EQ, LI Drawn 1 lesson Individual instruction by teacher 8n + 11 = 5n + 50 Grade 7; 6 students Descriptive; case studies with BM   • BM is suitable for demonstrating cancelation of identical terms on both sides of the eq.
• BM is not suitable for modeling equations with subtraction
Mann (2004) EQ Physical, drawn   Classroom instruction by teacher ■ = ■ ■ 5 + 6 = __ + 2 Grade 3; 1 class
Marschall and Andrews (2015) EQ, LI Drawn   Classroom instruction by teacher x + 1 = 3 4x − 3 = 2x + 5 Grade 6; 6 classes
Ngu, Chung, and Yeung (2015)   Drawn 40-min lesson Individual instruction sheet with BM 5 + 3n = 10 3m − 1 = 5 Grade 8; 71 students Pre-posttest; BM-group and comparison group CG: solving equations with inverse operations • BM-group improved from pre- to posttest
• CG improved more than BM-group
• Higher cognitive load for BM-group than CG
Ngu and Phan (2016)   Drawn 45-min lesson Individual instruction sheet with BM n / 2 = 7 x − 9 = 4 Grade 7; 63 students Pre-posttest; BM-group and comparison group CG: solving equations with inverse operations • BM-group improved from pre- to posttest
• CG improved more than BM-group
• Positive relation between performance on procedural knowledge and performance on conceptual knowledge for CG but not for BM-group
Ngu, Phan, Yeung, and Chung (2018)   Drawn Two 40-min lessons Individual instruction sheet with BM 3x + 1 = 2x + 8 6 − q = 10 Grades 8–9; 29 students Pre-posttest; BM-group and comparison group CG: solving equations with inverse operations • BM-group improved from pre- to posttest
• CG improved more than BM-group
• Higher cognitive load for BM-group than CG
Orlov (1971) PE Physical 2 years Classroom instruction by teacher 5x − x + 2 = 2x + 6 Grade 8; 200 students Repeated measures; BM-group and comparison group CG: experimental program without BM • BM-group, especially average and above-average students, outperformed CG
Perry, Berch, and Singleton (1995) PE Physical 1 lesson Individual instruction by teacher 3 + 4 + 5 = __ + 5 Grades 4–5; 56 students Pre-posttest; BM-group and comparison group CG: only verbal instruction • BM-group outperformed CG
Raymond and Leinenbach (2000)   Drawn 26 lessons Classroom instruction by teacher x + 4 = 2x + 3 Grade 8; 120 students Descriptive; BM-group   • BM instruction leads to better performance than textbook instruction
• Large performance decrease when returning to textbook after BM
• Better than expected performances on standardized algebra test after BM
Rystedt, Helenius, and Kilhamn (2016) EQ Drawn 1 lesson Classroom instruction by teacher 4x + 4 = 2x + 8 Grades 6–7; five classes
Smith (1985)   Physical   In pairs with BM 8w = 120 Grades 4–6 Descriptive; BM-group   • BM assisted in exploring/learning basic algebraic principles and enhanced motivation
Suh and Moyer (2007) PE, MR Group 1: Virtual Group 2: Drawn 5 lessons Classroom instruction by teacher; students individually with BM 2x + 2 = 10 Grade 3; 36 students Pre-posttest; two BM-groups   • Both BM-groups improved
• Each of the BMs showed unique features to support learning
Taylor-Cox (2003) EQ, PE Physical 1 lesson Classroom instruction by teacher A + C + B = C + A + B Grade 1
Vlassis (2002) EQ, LI Drawn 16 lessons Classroom instruction by teacher 7x + 38 = 3x + 74 13x − 24 = 8x + 76 Grade 8; 40 students Descriptive; BM-group   • Balance method was used by all students
• After BM instruction, students made many mistakes related to negative numbers and unknowns
Warren and Cooper (2005) EQ, MR Physical; drawn 4 lessons Classroom instruction by teacher ? + 7 = 11 ? − 4 = 13 Grade 3; 20 students Descriptive; BM-group   • Most students could represent equations with the BM and translate the model into symbolic eqs.
• BM assisted students in understanding the equal sign and solving for unknowns
• Ten students used the balance method for solving a subtraction problem; for others further teaching was necessary
Warren and Cooper (2009) EQ, PE, MR Physical; drawn 5 years Classroom instruction by teacher ? + 2 = 5 ? − 3 = 6 Grades 2–6; 220–270 students; Descriptive; BM-group   • BM enhanced understanding of language and symbols
• Students could generalize balance method for simple equations
• Older students could generalize the balance method for all operations
Warren, Mollinson, and Oestrich (2009) EQ, MR Physical, drawn   Classroom instruction by teacher 5 + 1 = 2 + 4 Kindergarten
1. Empty cells indicate that this information was not provided in that article
2. aBM Balance model
3. bEQ = rationales related to the equality aspect, PE = rationales related to the physical experiences, MR = rationales related to learning through models and representations, LI = limitation of using the balance model
4. cInformation about the research design was only included for articles in which the effect of the balance model on students’ learning outcomes was evaluated
5. d“Balance method” refers to the method of solving an equation by performing the same operations on both sides of the equation