- Title: Exploring how understanding of the equals sign relates to additive and multiplicative equivalence performance beyond arithmetic fluency
- Authors: Phoebe Mills et al.
- Access the original paper here
- Watch a video overview:
Paper summary
This preregistered study looked at how 277 final-year primary pupils in Northern Ireland understand the equals sign, and whether that understanding matters once you account for how quickly and accurately they calculate. Pupils defined the symbol themselves, rated fictitious pupils’ definitions, and completed timed tasks on standard calculations and on equations where the equals sign sits in an unusual place (such as 10 + 1 = 9 + __). Almost all pupils (96.4%) could name the symbol. Nearly half described it as a signal to work out an answer, 22% said it shows both sides have the same value, and none said the two sides could be swapped. Calculation skill explained most differences in performance, but pupils who saw the equals sign as meaning “the same value” still did better on both addition and multiplication equations.
If teachers remember one thing from this study, it should be…
Most pupils leaving primary school still read the equals sign as “the answer comes next”. Those who understood it as meaning both sides have the same value did better on equations with the answer in an unexpected place, even after allowing for calculation skill. That meaning needs teaching explicitly.
*** PAPER DEEP DIVE ***
What are the key technical terms used in the paper?
- Arithmetic fluency: speed and accuracy in standard calculation.
- Equivalence problems: equations where the answer doesn’t directly follow “=” (e.g. 10 + 1 = 9 + __).
- Operational view: “=” means calculate.
- Sameness view: “=” means both sides have equal value.
- Substitution view: “=” means either side can replace the other.
What are the characteristics of the participants in the study?
277 Primary 7 pupils (final year of primary school) from 11 schools in Belfast and surrounding areas of Northern Ireland. Their average age was 11.4 years and 48% were girls. Free school meal eligibility across the schools ranged from 4% to 74%. Pupils were tested once, without any intervention.
What does this paper add to the current field of research?
The link between understanding the equals sign and solving equations is well documented in younger children, largely in US studies. This UK study shows the link holds for 11-year-olds even after accounting for calculation skill, and extends it to multiplication equations, where the connection was roughly twice as strong.
What are the key implications for teachers in the classroom?
- Find out what your pupils think the equals sign means. Nearly every pupil could name the symbol, yet only around one in five described it as showing two sides of equal value. A further fifth just wrote “equals” or “equals to”, which tells you very little. At the start of a unit, ask pupils to write down what “=” means, and when you get a vague or answer-focused reply, follow up with “Can it mean anything else?” If you teach Year 7, it is safe to assume that many new arrivals still think of it as “the answer goes here”.
- Use the language of sameness when you read equations aloud. The authors argue that pupils rarely hear the equals sign described as a relationship, and most rated “work out the result” as the cleverest definition. Small changes in how you talk help here. Say “three add four is the same as seven” or “has the same value as” rather than “makes” or “gives you”, and encourage pupils to use that phrasing when they explain their own working.
- Move the equals sign around. Pupils made the most mistakes when calculations appeared on both sides, and most of those mistakes came from calculating in the wrong direction. Given 5 + 12 = __ + 3, a pupil might write 17 (adding the numbers before the sign) or 20 (adding everything). Mix formats in practice sets, such as 7 = 3 + 4, 6 + __ = 7 and 10 + 1 = 9 + __, and put one of those wrong answers on the board for the class to discuss why it doesn’t work.
- Give multiplication and division the same treatment. Most previous research, and plenty of classroom practice, focuses on addition. In this study, seeing the equals sign as “same value” explained about twice as much of the variation in multiplication equations. Include problems like 5 × 6 = 10 × __ and 12 ÷ 3 = __ × 2 alongside the addition examples, and ask pupils to check that both sides really do come to the same amount.
- Keep working on number facts as well. Calculation skill accounted for over 70% of the differences in how pupils did on these equations, far more than their understanding of the symbol. Pupils who recall facts quickly have more attention to spare for the structure of the equation. So keep regular fact practice going alongside the work on what “=” means.
Why might teachers exercise caution before applying these findings in their classroom?
The study is correlational and tested pupils once, so it can’t show that teaching the sameness meaning causes better performance. The extra variation explained was small (1–2%). All pupils were from Northern Ireland, and the timed tasks meant few pupils reached the hardest items.
What is a single quote that summarises the key findings from the paper?
“These findings suggest that while arithmetic proficiency is important for success on equivalence problems, knowledge of the relational meaning of the equals sign also plays a role, particularly in more complex equivalence contexts.”
