^{a}

^{a}

^{b}

^{a}

Students’ conceptions of the equals sign are related to algebraic success. Research has identified two common conceptions held by children: operational and relational. The latter has been widely operationalised in terms of the sameness of the values on each side of the equals sign, but it has been recently argued that the substitution component of relational equivalence should also be operationalised (Jones, Inglis, Gilmore, & Dowens, 2012, https://doi.org/10.1016/j.jecp.2012.05.003). In this study, we investigated whether students’ endorsement of the substitution definition of the equals sign is a unique predictor of their algebra performance independent of the other two definitions (operational and sameness). Secondary school students were asked to rate the ‘cleverness’ of operational, sameness, and substitution definitions of the equals sign and completed an algebra test. Our findings demonstrate that endorsement of substitution plays a unique role in explaining secondary school students’ algebra performance above and beyond school year and the other definitions. These findings contribute new insights into how students’ algebra learning relates to their conceptions of the equals sign.

Algebra is a central topic within school mathematics (

There are large differences between individuals in the way they solve and perform on algebra questions (e.g.

In most research, students’ conceptions of the equals sign (=) have been investigated in two ways: as an operator indicating the result of an arithmetic operation, and as a relational symbol indicating that both sides of the equals sign are the same and interchangeable (e.g.

Many students hold an operational conception of the equals sign, i.e., they view it as a ‘place-indicator’ for the result of an operation (

The general consensus among researchers, who investigate the relation between students’ understanding of the equals sign and their success at further mathematics, is that developing a relational conception of the equals sign, and moving on from the basic operational conception, is important for learning algebra (

Although these previous studies have defined relational definitions as “equal and interchangeable” the researchers have tended to focus exclusively on the sameness component in their operationalization of the definition, likely because children rarely spontaneously provide definitions that invoke the interchangeability of the two sides of an equation. Thus, previous work has led the field to assume that developing the sameness conception of the equals sign, and moving on from the operational conception, are assumed to be central to algebra performance.

The substitution conception arises from the formal mathematical definition of equivalence. If a relation is binary, reflective, transitive and symmetrical, it is an equivalence relation. The substitution conception is based on transitivity and symmetry properties of equivalence. Transitivity means that if

The substitution component of equivalence tends not to have been operationalised by education and cognitive researchers (see

Solving simultaneous equations can necessitate an understanding of the mathematical notion of substitution. Secondary-school students are usually taught to solve pairs of equations involving two unknowns such as Items 8 and 9 depicted in the instrument (see the Supplementary Materials section). A typical method to solve such problems is to express one unknown in terms of another, and then substitute that expression into an equation. However, as evidenced in several studies, students often struggle to substitute one expression for another, even in the later years of secondary schooling (

Despite the large body of research investigating the role of the sameness and operational conceptions of the equals sign in learning algebra, the role of substitution on students’ algebra performance is yet unclear. To fill this gap in the literature, the present study examined the role of substitution on algebra performance. On the basis of the existing literature, we hypothesised that:

On average secondary students would endorse the sameness definition more highly than the substitution definition. This hypothesis was based on previous work using similar instruments (

Endorsement of substitution would be a unique predictor of secondary students’ algebra performance independently of operation and sameness. This hypothesis was based on the finding that Chinese students, who outperform English students on algebra items, rated substitution more highly than their English peers.

A total of 57 secondary school students (_{age} = 15.34 years, age range = 14-16 years,

All students gave written consent to participate in the study. The mathematics teacher of each class was informed about the purpose and procedure of this study. Study procedures were approved by the Loughborough University Ethics Committee.

The students were given an instrument, which included an equals sign test and an algebra test (see the Supplementary Materials section). The instrument was completed by the students within their usual lesson time under test conditions and administered by their regular mathematics teachers. The students worked individually through the instrument and were allowed 20 minutes to complete all 11 items.

The equals sign test (Items 1 and 2) consisted of a subset of items based on

For each student, substitution, sameness, and operation scores were calculated through adding 1 point to the score for Item 1 if the student chose the related conception as the best definition of the equals sign for Item 2. For instance, if a student rated ‘= means the same as’ as ‘very clever’ (2 points) and chose this definition as the best, their sameness score was 3 (2 + 1) or if a student rated ‘= means the two sides can be swapped’ as ‘sort of clever’ (1 point) and chose another definition as the best, their substitution score was 1 (1 + 0). Each conception score for each student ranged from 0 to 3.

We used only closed items in which students rated definitions of the equals sign since students tend not to offer substitution definitions to free response items asking them to define the symbol ‘=’, although other authors have used free response items (e.g.

The algebra test was designed to explore students’ performance on solving algebraic questions. We adapted nine items from existing instruments (

Three items adapted from

Students were asked to solve linear and simultaneous equations in Items 6 to 9. These were based on typical items from high-stakes national examinations (General Certificate of Secondary Education) sat by most students in England at age 16. Finally, Items 10 and 11 asked students to assess the mathematical equivalence of equations consisting of letters. These last two questions were taken from the Increasing Competence and Confidence in Algebra and Multiplicative Structures (ICCAMS) project survey (see

We checked the performance of the items in the algebra test in terms of the internal consistency. Items 3, 5, and 6 were excluded from the analysis due to ceiling effects (see

Question | Min | Max | |||
---|---|---|---|---|---|

3A | 57 | 0 | 1 | 0.95 | 0.225 |

3B | 57 | 0 | 1 | 0.96 | 0.186 |

4 | 57 | 0 | 2 | 1.79 | 0.491 |

5 | 57 | 0 | 1 | 0.96 | 0.186 |

6 | 57 | 0 | 1 | 0.98 | 0.132 |

7 | 57 | 0 | 1 | 0.86 | 0.350 |

8 | 57 | 0 | 1 | 0.79 | 0.411 |

9 | 57 | 0 | 1 | 0.26 | 0.444 |

10 | 57 | 0 | 1 | 0.54 | 0.503 |

11A | 57 | 0 | 1 | 0.82 | 0.384 |

11B | 57 | 0 | 1 | 0.32 | 0.469 |

11C | 57 | 0 | 1 | 0.40 | 0.495 |

11D | 57 | 0 | 1 | 0.19 | 0.398 |

Students’ responses to the algebra test were scored for correctness irrespective of the solution method (i.e., 1 for a correct answer, and 0 for an incorrect answer). For each student, total scores were calculated, which we call here ‘algebra performance’.

Descriptive statistics for all measures are displayed in ^{i} As shown in

Variable | Min | Max | ||
---|---|---|---|---|

Algebra performance | 5.98 | 2.45 | 0 | 10 |

Substitution | 0.95 | 0.86 | 0 | 3 |

Sameness | 2.18 | 0.98 | 0 | 3 |

Operation | 1.77 | 0.85 | 0 | 3 |

Variable | 1 | 2 | 3 | 4 |
---|---|---|---|---|

1. Algebra performance | – | |||

2. School year | .76** | – | ||

3. Substitution | .46** | .35** | – | |

4. Operation | -.34** | -.55** | -.07 | – |

5. Sameness | .38** | .35** | .09 | -.55** |

**

Since school year, operation, sameness, and substitution were significantly correlated with the dependent measure, all of these variables were used as predictors for algebra performance in a simultaneous multiple regression analysis. In the multiple regression model, we entered all variables in the same step to test our hypothesis.

Regression results showed that the model was significant, ^{2} = .63, which meant that our model explained 63% of the variance in algebra performance.

Predictor | Β | η^{2}_{p} |
95% CI | |
---|---|---|---|---|

Model for All Items (Total ^{2} = .66) |
||||

Constant | 1.48 (1.03) | [-0.59, 3.56] | ||

School year | .72** | .48 | 3.50 (0.51) | [2.48, 4.53] |

Operation | .19 | .05 | 0.55 (0.32) | [-0.10, 1.19] |

Sameness | .21* | .08 | 0.53 (0.24) | [0.04, 1.02] |

Substitution | .20* | .09 | 0.56 (0.27) | [0.07, 1.05] |

Model for Typical Items (Total ^{2} = .47) |
||||

Constant | 0.14 (0.45) | [-0.77, 1.06] | ||

School year | .76** | .41 | 1.35 (0.22) | [0.90, 1.80] |

Operation | .32* | .10 | 0.37 (0.14) | [0.05, 0.62] |

Sameness | .26* | .08 | 0.23 (0.11) | [0.02, 0.44] |

Substitution | .05 | .00 | 0.05 (0.11) | [-0.26, 0.17] |

Model for Conceptual Items (Total ^{2} = .49) |
||||

Constant | 0.24 (0.81) | [-1.40, 1.87] | ||

School year | .60** | .31 | 1.96 (0.40) | [1.15, 2.76] |

Operation | .07 | .00 | 0.13 (0.25) | [-0.38, 0.63] |

Sameness | .10 | .02 | 0.17 (0.19) | [-0.21, 0.56] |

Substitution | .23* | .09 | 0.43 (0.19) | [0.05, 0.82] |

*

The algebra test contained items that are typical of those used in national examinations (Items 7 to 9) and items sourced from a conceptual test (

For the typical items the regression model was significant, ^{2} = .47, explaining 47% of the variance in scores. The regression coefficients are shown in ^{2} = .49, explaining 49% of the variance in scores. As shown in

In summary, operation and sameness were significant predictors of performance on those algebraic items that are typical of national examinations, whereas only substitution was a significant predictor of performance on items designed to test conceptual knowledge of algebra. We consider the implications of this in the discussion section.

To explore developmental issues, albeit tentatively, we considered the extent to which patterns of ‘accepting’ or ‘rejecting’ each definition relates to algebra performance in our data.^{ii} The full results are shown in

Operation | Sameness | Substitution | ||
---|---|---|---|---|

R | R | R | 1 | 6.00 |

R | A | R | 19 | 6.32 |

R | A | A | 6 | 8.50 |

A | R | R | 12 | 3.83 |

A | R | A | 2 | 8.00 |

A | A | R | 13 | 5.69 |

A | A | A | 4 | 7.00 |

The findings of the present study shed light on the role that substitution plays in students’ algebra performance. Multiple regression analysis demonstrated that endorsement of substitution was a unique predictor of algebra performance independent of operation, sameness and school year. This novel finding has important implications for educational assessment and practice; it indicates that enhancement of substitution could potentially influence students’ success in algebra. Future research should further examine whether, through training, developing a substitution conception of the equals sign can improve students’ algebraic learning and problem solving.

We found a positive correlation between algebra performance and sameness, and a negative correlation between algebra performance and operation. These results were expected because it has been evidenced that students who mainly endorse a sameness conception perform better in algebra than those who view the equals sign operationally (

However, for the three typical Items (7 to 9), which were based on national examination (GCSE) items, only operation and sameness were significant predictors, whereas for the three conceptual items (10 and 11), only substitution was a significant predictor. We are unsure as to the explanation for this but a possibility is that the typical items lend themselves to being solved using arithmetic methods. For example, Item 7 can be completed by guessing values for

Our finding that endorsement of the concept of substitution uniquely predicted algebra performance overall and on the conceptual items further highlights that substitution is an important conception of the equals sign which relates to students’ algebra learning. Given that

We also replicated previous findings showing that English students rated the sameness definition of the equals sign more highly than the substitution definition (

In conclusion, the present study contributes new insights into how students’ algebra learning relates to their conceptions of the equals sign. Previous research operationalised a sophisticated understanding of the equals sign in terms of sameness and tended to neglect substitution. Our findings show that the operation, sameness, and substitution conceptions of the equals sign are different constructs and substitution is a unique predictor of algebra performance independent of operation and sameness. Designers of both classroom instruction and of instruments designed to investigate students’ conceptual development should consider incorporating the equals sign in terms of substitution.

The authors would like to thank all the participating students.

This result holds using a non-parametric paired Wilcox signed rank test, Z = 1123,

We are grateful to a reviewer of an early draft of the manuscript for this suggestion.

The authors have no funding to report.

The authors have declared that no competing interests exist.