From: The balance model for teaching linear equations: a systematic literature review

Article |
Rationale^{b}
| Appearance | Use | Students involved |
Research design^{c}
| 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; BM^{a}-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 method^{d} 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 |