Not all uses of equals signs are equal
Resources in maths and stats for a pandemic
18 March 2020Multiplication Fact Fluency
8 March 2021The problem with equals signs
The sign “=” was invented in 1557 by Robert Recorde a Welsh mathematician. He was tired of having to write out the phrase “is equal to” too often. Now we cannot imagine maths without an = sign.
Using an equals sign correctly can be a challenge. Whenever an equals sign is used, it signifies that the expressions on either side are equal. A sequence of expressions separated by equals signs should all be equal to each other. For example: 4 + 5 = 3 × 3 = 21 – 12
However, the equals sign often gets used in an operational way, meaning “give the answer”. Our calculators do not help at all, as we press the “=” sign as an instruction to calculate. Many students have an operational view of the equals sign. The problem with this is that the operational use of the equals signs leads to incorrect equality statements.
The running equals sign
How many mathematics teachers have felt their life-force draining out on reading the following:
2 + 3 = 5 × 2 = 10 – 9 = 1
The person has added two and three to get five, which they then multiply by two. Next they subtract nine to get end up with 1. The whole procedure could have been written (2 + 3) × 2 – 9
Or it could have been written 2 + 3 = 5 5 × 2 = 10 10 – 9 = 1
Either of the correct ways seems long and tricky compared with the misuse of the equals sign, sometimes called the running equals sign. My experience is that most students will do anything to avoid any more writing than absolutely necessary, which makes correctness and rigour even less attractive.
Things that are understandable as part of a narrative do not survive the permanent written form. You can see this problem in the TV programme, Countdown, where the maths person shows calculations as they are happening in narrative. Because they are explaining in real time, it seems more acceptable. And they use a horizontal line which, though it represents an equals sign, seems to be more appropriate with the narrative approach.
Spreadsheets contribute to the confusion, with the equals sign indicating that we are putting a formula in a cell. In programming an equals sign sometimes means “Check to see if these things are equal” and sometimes means assign the value of this expression to the variable.
As a mathematician I would love the equals sign to be used correctly. As a teacher of mathematics, I am finding this uphill work.
Correct understanding of the equals sign matters
Mathematicians work with other mathematicians. Mathematicians also come back to work they have done previously. It is vital that what is on the printed page is correctly written to enable logic and computation to be checked. This is especially true in education as teachers and instructors need to be able to understand the students’ thought processes in order to identify where mistakes are occurring.
In algebra and other higher level mathematics it is vital to understand the relational nature of the equals sign. Much of rearrangement and solving of equations relies on the correct use of the equals sign. I also wonder if the difficulty students have with inequality signs is related to misunderstanding of the equals sign.
What is the solution?
Teach correct use of the equals sign explicitly
Get students to find the mistakes in a theoretical student’s work. Put something like
25 + 30 + 27 = 82/2 = 41 on the board and get students to say whether or not it is correct, how to correct it and what they would say to help the person who wrote it to understand the mistake.
Talk about the operational usage of the = sign as being like pressing the button on the calculator to find the answer.
Avoid the tradition that “the answer follows the equal sign”.
I have been known to use writing the equal sign after an expression as an indication to students to give me an answer. Knowing better I would not do this again. I hope.
Model correct behaviour
A teacher should always show correct working in examples, making explicit why they are doing so.
Say “is equal to” rather than “makes”
Exercises with non-standard positioning of the = sign.
Here are some examples from the Equals Sign test, an instrument used to evaluate students’ understanding of the = sign. You can read more about it in the paper referenced at the end of this post.
At Grade 2 (Year 3) level
- _ + 3 = 5 + 7 =
- 4 + _ = 5
- 3 + 5 = 4 + _
At Grade 6 (Year 7) level
- 15 – 7 = _ + 5
- 6 + 3 + 7 = 5 + _
- 6 × _ = 40 – _
By providing questions that place the = sign in different positions we are helping to de-emphasise the operational use of the equals sign. I would also suggest exercises that have more than one = sign to draw attention to the ongoing equality across several expressions. If you are using a textbook check to see if there is variety in presentation.
Different symbols or conventions (just dreaming here)
The lack of an easily written and understood alternative to the equals sign makes it difficult to be correct without being too complicated.
I saw in a post from another country (Denmark perhaps?) that people cross out the equals sign when it ceases to be true. The equal sign was true when we wrote it, but because we are adding another step it is no longer equal. The example we have been using would look like this when it was completed.
2 + 3 ≠ 5 × 2 ≠ 10 – 9 = 1.
I suspect this would increase confusion unless it became a convention in a whole school system. Perhaps drawing a box around each completed step would be an interim solution.
Can we invent a new symbol? Preferably one that already appears on the keyboard? What about a hash? It’s like two equals signs across each other and could mean a temporary equals sign that has been superceded:
2 + 3 # 5 × 2 # 10 – 9 = 1
Or maybe 2 + 3 =≠ 5 × 2 =≠ 10 – 9 = 1
=≠ means the expressions used to be equal, but are no longer because we have carried on with other operations.
Or maybe » which points to where we are going and has two parallel bent lines like an equals sign.
(Note that these are just to stimulate thought – though Robert Recorde was the first person to use the equals sign – maybe we can invent the operate sign.)
Further reading
This is a useful reference that discusses the problem of the misunderstanding of the equals sign and reports research on students and textbooks from four different countries.
Your thoughts
This is a problem which mathematics teachers have grappled with for decades, possibly even centuries. I’d love to hear your suggestions on
- How to teach to avoid the problem of the running equals sign.
- What other symbols or conventions we have available or could invent.
12 Comments
I teach using a balance – the equals sign represents the fulcrum in the middle. This physically demonstrates the concept of equal to. I find that particularly useful when working on solving linear equations and stressing the importance of doing the same operation to both sides. Once the students see 3 blocks being taken off each side, that magic aha moment happens and they get it!
Nice example. How do you stop them going back to their old ways?
Having the equipment available and plenty of practical practice seems to stop them going back. The problem starts when they’re little and “the answer” is what goes after =. We need to change that terminology. Get rid of answer. Use is the same as instead. Consistently, right throughout school from year 0!
how do you do a balance with two fulcra?
Not easily, but you can replace one side with the next expression perhaps.
Thought from previous post. One of the objections to the HP RPN calculators was the lack of an
= sign. That was one of the strengths of that now obsolete system in most schools. It didn’t
have an = sign because it wasn’t needed! Thank you T I, Sharp, Casio and all of the other parallel bar machines for helping a poor choice on naming to eliminate a far superior system.
Yes, the = sign has had a challenging time on calculators. My Casio does not have an = sign, but rather EXE.
I like the phrase,” is equals to” . I will start using this word instead of makes it. Makes it sound more interesting or something worth checking out.
That is great to hear. It is tricky to change habits sometimes.
The mathematical phrase,” is equals to “, is more informative than makes it. Will start saying that instead of makes it.
I still use my old HP RPN calculator, and use an app that looks just like it on my cell phone.
A very important article. Thanks! Our words and conventions are so influencial. I also find it helpful to show that a number sentence can be read either way if it is equal. ie the ‘answer’ is not necessarily on the right and as you pointed out ‘non standard’ position of the equal sign helps students be flexible yet still accurate. This flexibility is needed when rearranging equations in algebra. Never too early to introduce to students.