# The Equal Symbol

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I taught my first math class in 1965.  I taught my last class in 2017.  It wasn’t until my penultimate year of teaching that I realized many students misunderstood the equal = symbol. The earliest cited paper, which I have seen, addressing the topic is dated 1974.  It seems this has been a problem for my entire teaching career.  Evidently, I should have been more aware of my students’ difficulties. All my reading and studying of cognitive science overlooked the studies and papers related to the = symbol.  I am sincerely sorry for not observing and addressing the problem earlier.  As a ”Johnny-come-lately” I intend to write a series (10 or more) of posts addressing the = symbol, difficulty for learners, possible causes, and solutions for these difficulties.

Let’s begin with a little history.

The = symbol was first used in 1557 by Robert Recorde, a Welsh mathematician, who tired of constantly writing the words “is equal to.”  He used the familiar = symbol of two parallel lines of equal length because “no two things can be more equal.”  It is important to note that right from the beginning and continuing without modification to the present time the = symbol means “is equal to.”  For a brief but engaging account of Recorde’s short life read https://arstechnica.com/science/2017/07/how-the-equals-sign-changed-the-world/.

The symbol = was not immediately popular. Other symbols were used by some, but eventually the = symbol prevailed and, in mathematics, has from 1557 meant “is equal to.”

Please observe that the introduction and all early use of the = symbol was related to mathematics.  Today the = symbol is used in other contexts with different meanings; for example in computer programming languages = usually means replacement.  I guess we should be careful to stipulate that in mathematics discourse the symbol = means “is equal to.”

We have just argued for the need for stipulative definitions.

Unfortunately, in recent history misunderstandings have arisen probably because of well meaning but incorrect instruction in the schools of the world. Yes, I said schools of the world.  Misunderstanding of the = symbol is prevalent in nearly every country of the world.  It seems that considerable study of the problem has taken place in France, The UK, and Australia.

“About 70 percent of middle grades students in the United States exhibit misconceptions, but nearly none of the international students in Korea and China have a misunderstanding about the equal sign, and Turkish students exhibited far less incidence of the misconception than the U.S. students,” note Robert M. Capraro and Mary Capraro of the Department of Teaching, Learning, and Culture at Texas A&M.

Students who correctly understand the equal sign are more successful in mathematics and fields that require mathematics like engineering, according to their research.

The equal sign is pervasive in mathematics beginning as early as kindergarten. My personal opinion is that a symbol like the equal sign =, which requires considerable abstract thought, should not be introduced at such an early age.  My knowledge does not allow me to suggest an age at which it is appropriate to introduce the = symbol and its related abstract reasoning, but I am sure kindergartners do not possess such reasoning ability.  Introducing the use of any symbols in kindergarten seems just plain silly.

## 2 thoughts on “The Equal Symbol”

1. Michael Bächtold says:

Interesting post! Could you provide pointers to those studies of students difficulties with equality? I agree that Kindergarden seems to early to introduce =, but on the other hand I wonder, since for Recorde the sign = was just an abbreviation for the words “is equal to”, would also the words “is equal to” to be something that should be avoided in early mathematical education?

Since even nowadays the meaning of equality is not completely undisputed in mathematics (consider Voevodsky’s recent Univalent Foundations Program https://www.math.ias.edu/vvmc2018/agenda#PD ) these seem like hard questions.

1. Michael;
I am working on two future posts which should address your observations. One is a summary of what I have learned and my conclusions. The other is a list of references.
I will have to look at the reference you sent before going down that road.
Del