## Thoughts on Equal Signs

I was reading a recent report that stated that 70% of US students do not understand the equal sign in math. Robert Talbert has a nice article on this on his Casting Out Nines blog – he has a great blog that I highly recommend.

I was at a meeting in which the use of the equal sign became a heated discussion. I thought it might come to blows, but fortunately lunch showed up just in time. The discussion focused on the use of equal signs when simplifying expressions, whether they are algebraic expressions or just arithmetic expressions. The standard way to simplify an expression is to “complete a step” on the line below the expression, and place an equal sign in front of it. This states that the above expression is equivalent to expression below. Some instructors felt it was OK to leave the equal sign out because by writing the new expression below it was implied that they were equivalent. Here’s what I mean.

$2(x-7)+4$
$2x-14+4$
$2x-10$

$2(x-7)+4$
$= 2x-14+4$
$= 2x-10$

The contention was that by forcing students to use the equal sign when simplifying expressions, they confuse the equal sign’s use within an equation. The most likely error this leads to is the insertion of an “$= 0$ at the end of a problem, which leads to students finding a solution to a non-existent equation when they were only supposed to simplify an expression.

Of course, if students understand what the equal sign represents in each context this will not happen. How do we get them to understand? By clearly explaining the use of the sign when we are teaching students how to simplify expressions, and again explaining the use of the sign in an equation. If you let them know it is important, they will make it a priority.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. Each Wednesday I post an article (or, apparently, more) related to teaching math on my blog. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.

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• 2. colintgraham  |  August 11, 2010 at 1:33 pm

This is an interesting point. Particularly since = only seems to have been introduced around the 1550s and took about 100 years or so before it became an accepted symbol as a replacement for “is equal to”. The fact that “is equal to” is different from “is identical to” with its own three-line symbol just complicates things further!

I found the most successful way of explaining the sign was to say that each side of the equation was on one pan of a balance, the fulcrum if you like. I said that each line of the = represented a plate on each side of the balance which had to contain the same “weight” to work… Some textbooks actually use this type of image too, so it’s not original to me!

Confusion can often arise too, if you write the equation as:
2 (x-7)+4 = 2x – 14 + 4 = 2x – 10
i.e. a three-part elision of two separate steps…
2 (x-7)+4 = 2x – 14 + 4
2x – 14 + 4 = 2x – 10

Perhaps the underlying problem is that the = is replaced by the word-string “is equal to” rather than “has the same value as”….
The = can also be used to assign a value to something, so:
“Let x = 1” is not conceptually the same as “If x = 1, then …”

We then can get in to arguments about equation vs. formula, but that is another blog post, I suspect!

Colin

• 3. georgewoodbury  |  August 11, 2010 at 7:07 pm

Thanks for sharing, Colin. You know, I had a colleague who had his students simplify expressions in paragraph form without using the symbol “=”.
For example:
2(x-7) + 4 is equivalent to the expression 2x – 14 + 4, which is equivalent to the expression 2x – 10.
His students never had trouble understanding the equal signs they used when solving equations, but I think they were shortchanged of the opportunity to determine for themselves the difference between different applications of “=”. They were also going to have trouble when they found themselves in another math class that used the equal sign when simplifying expressions.

By the way – great job with getting #mathchat up and running on Twitter. For those of you who haven’t participated, #mathchat is a math related chat on Twitter that takes place every Thursday at 23:30 GMT. Check it out!
George

• 4. colintgraham  |  August 13, 2010 at 3:47 am

Thanks for the #mathchat plug! We have a Wiki which has lots more information too: http://mathschat.wikispaces.com/

I’m not sure I would go so far as ‘paragraphing’ by I certainly see a value in oral expressions of equations, particularly if we are pandering to the ‘fetish’ of writing down every step to show working…

Would any of us accept 2 (x-7)+4 = x – 5 as being ‘correct’ in solving or simplification? Probably not, unless we asked the student to explain their steps or write them down. When I saw the first expression, I divided by 2 rather than multiplying out the parentheses and collecting common terms… But when you are ‘taught’ solution/simplification, starting off by dividing by 2 is not the ‘proper’ method, although it is one step more efficient!

I’m greatly in favour of getting students to explain out loud how they got their answers, because that can sometimes reveal different types of thinking or approach that are just as valid. The ‘proper’ algorithmic method has it’s place, but maybe only to get those extra marks in an examination…!

I also thought I should expand a little on the different concepts of =. As an example of verbalizing, how would you say:

“If x = 2, then 2x + 7 = 11”

If we say “If ex equals two, then two ex plus seven equals eleven,” then we are probably just assisting in the confusion!

My way/suggestion:
If ex is assigned a value of two, then two ex plus seven has a value of eleven.

• 5. elissamilne  |  August 11, 2010 at 6:06 pm

Really?!

I’m not a maths teacher, so I have nothing anecdotal to share, but I’m absolutely amazed that students have any trouble whatsoever with reading, understanding, using =.

It’s about one of the most useful signs there is – whether we are talking maths or twitter!

But I suppose what is deeply disturbing is that students who do not understand ‘=’ almost certainly have no understanding of ‘equivalence’. And a failure to be able to categorise things as being equivalent would result in citizens unable to make good choices in any number of aspects of their lives.

• 6. georgewoodbury  |  August 11, 2010 at 6:56 pm

Thanks for sharing, Elissa. I am equally disturbed.
I’m afraid that the problem may be that we focus on converting 2x – 14 + 4 to 2x – 10 without spending much time on the notation and the meaning of “=”. Some instructors simply state that you must put an equal sign on the beginning of a new line without helping students to understand what it means and why we do it. I suppose what I am trying to get at is that the focus must be on understanding, not being able to simply copy steps.
George

• 7. noemie  |  August 12, 2010 at 12:43 am

I’m a linguist and therefore it has been a while since I’ve had to complete algebraic equations, but I’m fairly sure when we were simplifying we didn’t put the equal sign at the start of the line (mainly I suspect because it was a full equation which already had a = in the middle of it). If you look at it from the linguistic point of view though, starting a new paragraph means a clear new start. I would have thought that it was clear in the student’s mind they were simplifying the exact same equation and that the = is implied in their mind, much the same as they might copy a paragraph in neat underneath what they’ve written in rough?

• 8. georgewoodbury  |  August 12, 2010 at 5:42 am

Good point – this was essentially the argument of those who felt it should be left out. “Why would we write a step on the next line that was not equivalent?” I think that many teachers leave the equal signs out, except when solving an equation such as 2x – 14 = 4. I do not think that it would be difficult to increase our focus on proper usage of equal signs, but I think that many of us do leave them out when simplifying an expression line by line.
George

• 9. Raymond Johnson  |  August 13, 2010 at 2:56 pm

To get my students thinking about the proper use of the equal sign, I’d suggest this guideline: “If a problem does not give you an = sign, think very carefully before deciding to insert one yourself.” That’s usually a good starting point for discussions about where the equal sign is and is not appropriate. For people surprised that equal signs are so easily misused, they’ve never seen how many students read this:

“Multiply 5 by 4, then add 7, then divide by 3”

and write this:

5 x 4 = 20 + 7 = 27 / 3 = 9

Which, of course, says that 20, 27, and 9 are all the same value.

And yes, I’m one of those teachers who does not use equal signs when simplifying. Even if I’m forced to write horizontally instead of vertically, I’ll use arrows to keep from confusing students.

• 10. Pete Horne  |  August 14, 2010 at 7:53 pm

I’ve faced the same dilemma regarding students turning a simplification into an equation. When I am demonstrating, I will tend to use an arrow, rather than an equal sign. This can be done vertically as well as horizontally, such as

2(x-10)
-> 2x – 20

It also helps to justify the step in the line above the arrow:

2(x-10) “Distribute”
-> 2x – 20

• 11. lordaxil  |  August 16, 2010 at 9:26 am

Why would you not use an = sign? The expressions above and below are indeed equal, and solving the resulting equation tells you that x can take any value which, although not very useful, is nevertheless true.

• 12. Whit Ford  |  August 15, 2010 at 12:42 pm

I see four potential issues at play here:

1) Notation. With every passing year I become more firmly convinced that many students don’t fully understand some of the notation they are using. Leading culprits include: 2x (multiplication despite no notational symbol), expressions written as fractions (it’s a division problem, but can be read as a fraction) and = (equivalence, substitutability).

2) Concepts. Are students following a procedure by rote, or do they really understand the concept of equivalent expressions (or equivalent equations, depending on the problem)? If they are following a procedure by rote, then the risk of improper equal sign use rises dramatically.

3) Efficiency. Most students seem to have been taught only the most efficient way to solve math problems in their past studies – less efficient solutions often seem to have lost points for not being as efficient (or worse yet, are labelled as “wrong”). Math work is perceived as being all about not wasting either paper or ink (which, in my opinion, constrains thinking). From this perspective, the equal sign at the beginning of each line is a waste of both ink and paper.

4) Conflicting practices/notations. If we write an equal sign at the beginning of each line of work when rewriting expressions, why don’t we also do it when rewriting equations?

This dilemma is nicely handled by using an arrow (read as “becomes”, if you like), perhaps even make it a vertical arrow => pointing from the previous line of work to the next. An arrow can be used with both expressions and equations… not that an equal sign would be incorrect here, but I would find it visually slightly confusing: having only one equal sign per line when working with equations lets the equal sign serve as the “balance point” in the equation. This visual interpretation does not apply/work when working with expressions and writing equal signs at the start of each line.

My two cents!

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