The equals sign read as ‘makes’: a misconception with consequences
Why many children read the equals sign as an instruction to calculate, which errors follow from it and how primary teachers can address it in lessons.
In brief: Many children read the equals sign as ‘and now comes the answer’. Faced with 3 + 4 = □ + 5, they write 7 or 12 instead of 2, consider 7 = 3 + 4 to be false and write chains of calculations such as 8 + 2 = 10 + 3 = 13. Teachers can address this by introducing the sign from the start as ‘is the same amount as’, working with a balance and showing equations in both directions and with calculations on both sides. For grades 1 to 4, Mazio Maths has Basiscamp worlds on patterns and on balance that can accompany this work.
Children meet the equals sign in their very first week of maths lessons, almost always in the form 3 + 4 = □. This easily gives rise to an idea that works in grades 1 and 2 and causes problems later. This article shows teachers how to recognise the misconception, how to ask follow-up questions and which materials help to address it.
Two ways of reading the equals sign
Maths education research distinguishes an operational understanding, in which the sign is read as an instruction to calculate, from a relational understanding, in which it expresses that both sides have the same value. The first is enough for tasks such as 3 + 4 = □. As soon as there is a calculation on both the left and the right, the second is needed.
How large the difference can be is shown by a small pilot study with 60 children in grade 4, published in the Beiträge zum Mathematikunterricht 2018. Tasks of the type 4 + 8 = □ were solved largely correctly by 92 per cent of the children, tasks of the type 4 + 8 = □ + 7 by only 10 per cent. The study is small, but the pattern matches earlier research cited there.
Typical errors and what lies behind them
| Task | Child’s answer | What the child is thinking | Correct |
|---|---|---|---|
| 3 + 4 = □ + 5 | 7 | Calculate the left side, put the answer in the first empty box. | 2 |
| 3 + 4 = □ + 5 | 12 | Add all the numbers together. | 2 |
| 9 + 6 = 10 + □ | 15 | The answer belongs after the equals sign. | 5 |
| 7 = 3 + 4 | ‘false’ | The answer has to be on the right. | true |
| 5 + 2 = 2 + 5 | ‘false’ | There is no answer on the right. | true |
| Method for 8 + 5 | 8 + 2 = 10 + 3 = 13 | The sign links the steps of a calculation. | 8 + 2 = 10, 10 + 3 = 13 |
The last row arises easily because it is a convenient way to write down a method. The expression, however, states that 8 + 2 equals 13.
The consequences reach far. Children who read the equals sign only as ‘makes’ have difficulties with missing-number problems, with using turnaround facts and neighbouring facts, and later with equations in secondary school.
Diagnostic questions to ask
- Ask about the meaning. ‘What does this sign mean?’ Answers such as ‘that’s where the answer goes’ show the operational view. ‘Both sides are the same amount’ shows the relational view.
- True or false. Present equations such as 7 = 3 + 4, 3 + 4 = 4 + 3 and 3 + 4 = 8, and ask the children to decide and give reasons.
- Missing numbers with reasons. ‘What goes in 3 + 4 = □ + 5? How did you check that it is right?’ Children with a relational view work out both sides and compare them.
- Look at the children’s own methods. Look in exercise books for chains of calculations with several equals signs. They show how the child uses the sign.
What helps in lessons
- Change the way you say it. Read the sign as ‘is the same amount as’ or ‘has the same value as’, together with the children and at the board.
- Balance and equilibrium. On a balance scale or number balance, place weights on the left and right so that it balances. Then write down the matching equation, for example 3 + 4 = 2 + 5.
- Equations in both directions. From the start, also write 7 = 3 + 4 and 7 = 5 + 2, so that the answer is not always on the right.
- Calculations on both sides. Connect turnaround facts and neighbouring facts with the equals sign: 6 + 7 = 7 + 6, 6 + 7 = 6 + 6 + 1.
- Record methods correctly. Write intermediate steps as separate equations or show them with arrows, and discuss the difference with the class.
- Include inequalities. Tasks with <, > and = show that the sign expresses a comparison.
How Mazio Maths can help
Mazio Maths is a learning platform for maths lessons in primary school. The content is organised by the five content areas of the German national education standards, including patterns and structures, and is designed for the two-year bands of grades 1/2 and 3/4.
Patterns and balance in the Basiscamp. Alongside the calculation worlds, the Basiscamp has worlds on patterns, balance and puzzles, designed for grades 1 to 4. On the shelf, choose the ‘Basiscamp’ area and set the year group under ‘Grade’. With ‘Open & play’ you can see a unit beforehand exactly as a child sees it; what you do there does not count as a result.
Lesson ideas in the Collection. The Collection contains lesson ideas with a goal, a sequence of phases and matching materials, from 10 minutes to a double lesson, which you can filter by grade and duration. ‘Everything on the board’ puts all the materials for an idea on the board at once.
When a child makes a mistake. The learning assistant Mazio helps directly within the task, with a question, with highlights in the picture and, if needed, with a given partial step. He does not give the solution; the child takes the next step on their own.
The class’s misconceptions. Once a week, Mazio Maths checks whether the class’s wrong answers point to a widespread misconception. The support card explains the error with an example, suggests steps for the class under ‘In lessons’ and proposes units and Knobelbox tasks under ‘Matching content for the board’. Under ‘Results’, ‘Library’ opens the common misconceptions, ordered by content area. Under ‘Patterns and structures’ you can see which entries are listed there.
An example from the classroom. A grade 2 class works with the balance scale for a week. On Monday, the children build balances in groups and write equations in both directions. On Tuesday, the teacher chooses the ‘Basiscamp’ area under ‘Teaching’ on the shelf, uses the plus button to put an exercise from the ‘Balance’ world on the class board and adds the note ‘Say while calculating: is the same amount as’. The children see the exercise on their devices straight away. The difficulty adapts to each child’s previous work. On Friday, the class checks together whether equations are true or false, and the teacher notes who still says ‘that’s where the answer goes’.
You will find further misconceptions in grades 1 and 2 on the page Maths in grades 1 and 2 (in German). All units by topic are in the library of all units. How the class evaluation works is explained in the article How Mazio Maths analyses the class’s errors in thinking. Mazio Maths is free for teachers: create a free account. Using it with a class is currently available to schools in Germany; if your school is in another country, registration records your interest.
Frequently asked questions
What do children typically write for 3 + 4 = □ + 5?
Often 7, because they work out the left side and write the answer in the empty box, and sometimes 12, because they add all the numbers. The correct answer is 2, because both sides must come to the same amount. Ask the child how they checked.
When should the equals sign be introduced relationally?
From the start. If children in grade 1 only see tasks of the form 3 + 4 = □, the reading ‘makes’ becomes entrenched. Equations in both directions and the balance belong in the very first weeks.
Are chains of calculations such as 8 + 2 = 10 + 3 = 13 really a problem?
Yes, because they are mathematically incorrect and reinforce the reading ‘makes’. Write intermediate steps as separate equations or with arrows. Children adopt the notation they see on the board.