Equality – Awakening to a Problem Part II

Share Cognitive Science Implications for Teaching Mathematics

I do not remember exactly what we did in the early grades when I was a child but I am quite sure we were not presented with pages of problems written in the form 3 + 5 =   .  Addition problems were for the most part presented in a vertical format and did not lead to misconceptions about the meaning of the = symbol.  I believe there were also problems stated as: “find the sum of 3 and 4” and problems that asked us to “illustrate addition of 3 and 5 on the number line”.  For multiplication the presentation was again in a vertical format and relied strictly on memorization of the multiplication tables.  I do recall students being corrected when they referred to these as “times tables”.  It seems to me that at a very early age division exercises were constructed and stated in such a manner as to demonstrate the relation between dividend, divisor, quotient, and remainder as expressed by the division algorithm.  Subtraction was poorly taught as nothing but “take away”.  I don’t think the = symbol was introduced at a very age and therefore did not lead to the misconceptions which are common today.  As a final comment let me point out that I am particularly fond of writing everything in a horizontal format but introducing that format too early might be a mistake.

I do remember quite clearly when I learned about solving equations.  I have been forever grateful to that instructor (Mr. Parkhill) for establishing a proper foundation upon which I built a career in mathematics.

When we were introduced to linear equations in one variable we were informed about equivalence of equations and methods for generating an equivalent equation from a given equation.  When we began writing solutions of linear equations we were required to write the following:

5x – 7 = 3x + 5
is equivalent to 2x – 7 = 5
is equivalent to 2x = 12
is equivalent to x = 6
The solution is 6

In our oral explanations of our work, I don’t recall speaking about moving things from one side of the equality to the other.  I believe I learned that nonsense in college.  After some period of time my high school instructor permitted us to write:

5x – 7 = 3x + 5
⇔ 2x – 7 = 5
⇔ 2x = 12
⇔ x = 6
The solution is 6

As a result of this strictly correct introduction it never occurred to me or anyone else in that class to say or write that two equations were equal.  Eventually we were permitted to write our work in the following common conventional form.  I dread to think of the punishment if any one of us said two of the equations were equal.

5x – 7 = 3x + 5
2x – 7 = 5
2x = 12
x = 6

There was no danger of confusing that series of steps with the following computational steps which contain statements of equality.

(5 + 3i)(2 – 4i)
= 10 – 20i + 6i – (12i2)
= 10 – 20i + 6i + 12
= 22 – 14i

For some background reading about current misunderstandings of the = symbol you might start with the following.


  1. http://www.sciencedaily.com/releases/2010/08/100810122200.htm
  2. Use Google to search for “Does Understanding the Equal Sign Matter? Evidence from Solving Equations.” by Eric J. Knuth and others in 2006.
  3. http://www.home-school.com/Articles/the-equal-sign-symbol-name-meaning.php
  4. http://www.ksl.com/?nid=960&sid=19985258#IZ3EGzUO4kC3D7ze.99

A final observation.  Students grow to adulthood.  The implication is that many adults have a fundamental misunderstanding, Consequently they fail to use the symbol or use it incorrectly.

3 thoughts on “Equality – Awakening to a Problem Part II

  1. To me the problem stems from the fact that many American educators don’t understand how mathematics has very specific definitions. For instance, one of my grandkids brought home a homework paper that treated fractions and decimals as two separate subjects instead of two ways of representing points in a number line especially between whole numbers. My grandkids easily pick up the concept of a number line. I don’t think many teachers really understand it. A professor at UC Berkeley wrote a great paper on this to teach teachers that because he feels the same as I do. And he teaches teachers

    1. Thanks for the thoughtful response Tom. I agree with you that too many educators do not understand the role and significance of definitions in mathematics. However that is not the only cause for misunderstanding the equal symbol. I am convince that problem begins with your point and then is compounded by many other actions in the classroom.
      Can you find and furnish a reference to the article by the Professor UC Berkley?

    2. Tom;
      If you are interested in some of my thoughts about the importance of definitions in mathematics you might want to read the following two pages (on the Page menu) of this blog:
      Mathematics Without Definitions
      Vocabulary Strategies for the Mathematics Classroom

      In case you missed them, blog posts on the following dates also address the role of definitions in mathematics (you can find them in the Archives):
      In June 2018: 14, 19, 21, 26, 28
      In July 2018: 3, 17, 190, 24

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