The other day a student wrote the following on his quiz paper.

(5 + 3i)(2 – 4i)

10 – 20i + 6i – (12i^{2})

10 – 20i + 6i + 12

= 22 – 14i

In an effort to correct his writing I corrected his work as follows

(5 + 3i)(2 – 4i)

= 10 – 20i + 6i – (12i^{2})

**=** 10 – 20i + 6i + 12

= 22 – 14i

I added these notes:

- If two expressions are equal, indicate that fact with the use of the = symbol.
- Just because two expressions are written beneath one another does not indicate any relationship between the two expressions except in the case of solving an equation or an inequality.

I also circled the one equal symbol he had provided and asked “Where did this come from?”

The basis for my actions were.

- I assumed the missing equality symbols was nothing but “lazy writing”
- I was truly baffled by the fact that the student felt no compulsion to indicate equality of any two expressions until the very last line.

Since that time I have read a number of papers about some of these errors with the = symbol and have concluded that my feedback to the student really did not address his problems.

It turns out that many (if not most) beginning students use and read the = symbol to mean “the answer is”. Their view of the = symbol is operational rather than the correct relational understanding of the symbol. So this student was simply telling me “the answer is 22 – 14i”.

What I read was that 10 – 20i + 6i + 12 and 22 – 14i represent the same complex number and that meant they had the same real components as well as the same complex components. Not at all what the student thought he had said.

The missing = symbols are a consequence of our instructional methods which are more directed at “getting the answer” than understanding.

We permit students (and everyone else) to write something like the following accepted convention when solving an equation.

5x – 7 = 3x + 5

2x – 7 = 5

2x = 12

x = 6

Almost every College Algebra student will claim incorrectly that the first and second equations are equal, the second and third equations are equal, and so on to the end. The student quite naturally develops the idea that if things are written beneath one another they are presumed to be equal. Consequently it makes good sense to write the following.

(5 + 3i)(2 – 4i)

10 – 20i + 6i – (12i^{2})

10 – 20i + 6i + 12

= 22 – 14i

These and other misconceptions exist because we as teachers permit them to exist. In fact what we do in our classrooms fosters and encourages such misconceptions. We teach students to write using certain conventions without insisting that they know what is meant by the convention.

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 like 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 early 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 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

Û2x – 7 = 5

Û2x = 12

Ûx = 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 – (12i^{2})

= 10 – 20i + 6i + 12

= 22 – 14i

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

- http://www.sciencedaily.com/releases/2010/08/100810122200.htm
- Use Google to search for “Does Understanding the Equal Sign Matter? Evidence from Solving Equations.” by Eric J. Knuth and others in 2006.
- http://www.home-school.com/Articles/the-equal-sign-symbol-name-meaning.php

February 6, 2014 at 12:51 PM |

Thank you for sharing. I also tend to make the type of comments you did, even though I felt there was more in it than just “lazy writing”. Now this makes much more sens to me 🙂

I remember also in 2nd (somewhere between last year of high-school and first year of college) having a math teacher who was really strict in the use of “if and only if” (i.e. you “is equivalent to”) between equivalent equations. Later, I was introduced to the fact that “” implies something stronger than “=>”. Many student use “=” instead of “=>” or “”. Worse is that when I insist on this point, it seems that they just see me as a picky teacher. You are absolutely right when you say that “These and other misconceptions exist because we as teachers permit them to exist. In fact what we do in our classrooms fosters and encourages such misconceptions. We teach students to write using certain conventions without insisting that they know what is meant by the convention.”…