It depends what the definition of "is" is.

In the recent issue of Mathematics Teaching in the Middle School, I found an insightful article about the EQUAL SIGN. Who knew?

The authors gave students the following questionnaire:

The following questions are about this statement: 3 + 4 = 7 What is the name of the red symbol? What does the symbol mean? Can the symbol mean anything else? If yes, please explain.

I guessed my students would tell me that symbol means, "the same as," or, "both sides of the equation have the same value." Boy, was I wrong!

The article explains that more than half of the middle schoolers surveyed say that the equal sign is "operational" -- that is, they say it means, "find the sum of 3 and 4," or, "the answer to the problem on the left." The equal sign is actually "relational" -- something like, "is the same as," or, "the numbers on both sides are the same amount."

Well, what about my 8th-grade Algebra I students? I gave this questionnaire to 24 of my Algebra students, and roughly two-thirds of my students gave operational answers such as,

• The symbol shows that the answer is next to it.
• The two #s before it, when put together by the proper action of the symbol between it, will equal the number coming after it.
• It means the question should be solved.
• It means the two numbers sun is something when put together.

WOW. Algebra I students... successful, bright Algebra I students. I persuaded one of my colleagues to survey his pre-Algebra 8th graders, and the percentages were EXACTLY the same. Two-thirds of the students misunderstand the equal sign.

Does this matter? Sure. The article presents great evidence, too. They asked students to identify the value of the variable in problems like, 5m + 2 = 52. Students with a proper [relational] understanding of the equal sign answered these questions correctly TWICE as often as other students. And this makes sense to me. If a student has a poor understanding of the equal sign, problems like the one above are nonsensical. Solving them becomes a dance of gimmicks and rules instead of internalizing relationships. Could correcting such a BASIC piece of information result in better student performance in increased understanding?

We want students to understand their mathematics -- not simply sleepwalk through rote algorithms. It is impossible to achieve this level of algebraic reasoning without understanding the relationship represented by the equal sign.

I'm shocked that I have allowed my students to escape such a basic understanding. It makes me wonder what other basic skills and definitions could REALLY benefit my students.

Links:

See this described at Improbable Research