# Students' understanding of the equal sign not equal, prof says

##### Aug 10, 2010

Taken very literally, not all students are created equal — especially in their math learning skills, say Texas A&M University researchers who have found that not fully understanding the "equal sign" in a math problem could be a key to why U.S. students underperform their peers from other countries in math.

"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.

They have been trying to evaluate the success of education through students' interpretation of the equal sign. They have published several articles on this topic, with the most recent one published in the February 2010 issue of the journal Psychological Reports.

Students who exhibit the correct understanding of the equal sign show the greatest achievement in mathematics and persist in fields that require mathematics proficiency like engineering, according to their research.

"The equal sign is pervasive and fundamentally linked to mathematics from kindergarten through upper-level calculus," Robert M. Capraro says. "The idea of symbols that convey relative meaning, such as the equal sign and "less than" and "greater than" signs, is complex and they serve as a precursor to ideas of variables, which also require the same level of abstract thinking."

The problem is students memorize procedures without fully understanding the mathematics, he notes.

"Students who have learned to memorize symbols and who have a limited understanding of the equal sign will tend to solve problems such as 4+3+2=( )+2 by adding the numbers on the left, and placing it in the parentheses, then add those terms and create another equal sign with the new answer," he explains. "So the work would look like 4+3+2=(9)+2=11.

"This response has been called a running equal sign — similar to how a calculator might work when the numbers and equal sign are entered as they appear in the sentence," he explains. "However, this understanding is incorrect. The correct solution makes both sides equal. So the understanding should be 4+3+2=(7)+2. Now both sides of the equal sign equal 9."

One cause of the problem might be the textbooks, the research shows.

The Texas A&M researchers examined textbooks in China and the United States and found "Chinese textbooks provided the best examples for students and that even the best U.S. textbooks, those sponsored by the National Science Foundation, were lacking relational examples about the equal sign."

Parents and teachers can help the . The two researchers suggest using mathematics manipulatives and encourage teachers "to read professional journals, become informed about the problem and modify their instruction."

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##### gunslingor1
2.3 / 5 (3) Aug 10, 2010
I agree its these misconceptions about the underlying principles of subjects that cause the problems, though the teacher that teach using this method are probably the most to blaim. There's no such thing as a bad follower, only a bad leader.

All that being said, what on earth are you all talking about?

"...4+3+2=( )+2...So the work would look like 4+3+2=(9)+2=11"

I understand what your trying to teach kids with this kind of math problem, to add in segments, but its totally the wrong way to do it.

The mathematical notation 4+3+2=( )+2 literally means 4+3+2=0+2 which would be an inequality. Mathematical notations should not be circumvented to teach a point, this is a copout. There is a perfectly good correct way to do it. And you wonder why children are confused? How can you expect them to fully understand the equals sign when your not even using correct mathematical statements. It only confuses them more, and myself an engineer. Here's how to teach the idea 4+3+2=(a+b)+2.
##### ainsworth50
3 / 5 (2) Aug 10, 2010
Are you serious?
##### gunslingor1
1 / 5 (2) Aug 10, 2010
Me? SURE!

I've had these problems many times. While in college, I was in Cal II or diffEQ or something similar, handling complex math. May roomate was training to be a restaurant manager, so he took pre-algebra or something. He always would ask for assistance in solving simple equations. Well, most of the time I could not help him without an hour of study to learn the teachers bizarre affectation of a proper notation.

I mean, the freaking article is saying students aren't understand the equals sign for some STrAnGE ReAson, yet they're demonstrating they also do not understand the darn equals sign. I mean, come on, mathematically 4+3+2=( )+2 means nothing. Equations are supposed to be universal, you're not supposed to learn new notations for every simple problem you come across.
##### dirk_bruere
5 / 5 (6) Aug 10, 2010
call it:
4+3+2 = x+2
What is x?
##### Richak28
1 / 5 (2) Aug 10, 2010
All it tells me is no one knows what they are
doing, therefore the ones they are teaching
doesn't know what they should be doing !!!
##### RobertKarlStonjek
not rated yet Aug 11, 2010
The student error is in considering the equation to be a sequence of instructions, thus the first part is solved (4+3+2=11) and then the next instruction in the sequence is considered (+2). Thus their logic is good but it is misapplied rather than being outright flawed.

It might be less confusing if, as others above have suggested, correct notation is used ie 4+3+2=x+2, Solve the equation for x ie 4+3+2-2=x therefore x=4+3=7
##### jsa09
3 / 5 (2) Aug 11, 2010
I am with you all on this one. I have always been distressed by these problems because of the incorrect notation. The empty parenthesis imply to some that perhaps they should have something in them but an empty parenthesis is already empty and there is no real reason for it to contain anything.

I agree the notation should read 4+3+2=x+2 solve for x.

empty parenthesis is just asking for trouble and you will be sure to get it.
##### gunslingor1
3 / 5 (1) Aug 11, 2010
THANK YOU THANK YOU THANK YOU!!!!

If an engineer can't directly solve a simple arithmatical equation using known notation, how can you expect a student to. I mean, notations are standardized for a reason.
##### Sazzle
2.3 / 5 (3) Aug 11, 2010
I realise that when us Brits 'allowed' the Americans to look after themselves back in 1783 we didn't give them a very good grounding in English, but I didn't realise we left them basically inept at Maths. How the **** did you guys get to the Moon - aim and fire?

Really Sorry, but the Engineer's post begged a sarcastic response. I'm expecting a big bite back now - Ouch! ;-)
##### tkjtkj
5 / 5 (1) Aug 11, 2010
call it:
4+3+2 = x+2
What is x?

well, thanks to earlier contributor, x = a+b , so, we have:
4+3+2=(x[=a+b])+2
simply.
##### gunslingor1
5 / 5 (1) Aug 11, 2010
lol,Sazzle:

I don't think its the brits fault, lol. But we do have a problem with math in the US. Not so much a problem for engineers, physics dudes and math guys; the problem is for the lower math levels, english/law/medical majors and such.

These teachers are screwing with mathematics to try and make it easier to teach simple math with the justification that non-math guys will never need the real notation.

I sincerely feel approaching teaching math in this fashion is a huge mistake. They are putting a brick wall in front of these guys, if they do ever decide to learn more math. Plus, it makes it impossible for anyone who knows math to actually help them.
##### Blicker
1 / 5 (1) Aug 11, 2010
Once again, we all can see immediately where the fault lies; why can't they?
##### Sazzle
5 / 5 (1) Aug 11, 2010
I remember as a young girl in Junior school being taught that the = sign is like a pivot on a weighing scale and that bost sides have to balance. Never had a problem with it. I find maths far more beautiful than any English Poem. Its the means to describe the universe at all scales and, in its self, its a universal language. Such a shame so many people are mis-educated from the start; on both sides of the pond I may add. (Then again when I was at school we were taught from a basic grounding which, at the time was called "the three R's" ironically the three R's was a very bad name for a brilliant concept... Reading wRiting and aRithmetic... doh! some education authority 'tagline' c**p - it took me longer to work that out than it did for me to learn basic algebra). im rambling, sorry x
##### Eric_B
5 / 5 (1) Aug 11, 2010
the problem with math in the usa is that we only know how to count in terms of 3 (strikes/outs), 4 (downs), 9 (innings), 100 (yards) and 2000 (calories [of junk food])
##### gunslingor1
not rated yet Aug 11, 2010
I agree with the three of you, especially Sazzle (good writeup).

I remember my highschool chemistry teacher said a few things that really angered me; I challenged him and got suspended for this. Here's what he said:

1. "Global warming is largely caused by gas excaping from your tank while your fueling your car, so put the cap back on as quickly as you can."
-LOL, rediculous!
2. "Stomach acid can eat through a steak in 8 seconds, so why can't it eat through your stomach? The reasons is because stomach acid is diluted"
-No, it's because of polypeptides!

I only had a few intelligent teachers, the rest were practically incompetent.
##### Cyberguy
5 / 5 (1) Aug 15, 2010
@gunslingor1
Obviously the ( ) is supposed to represent an empty space where a number is to go. This is used in the very early stages before a child learns about x or y as variables - about the age of 6-8. In textboxs it is sometimes written as an empty square or an underscore instead of brackets. It is not a "notation" as such, just a teaching stage where the child tries to fill in the gaps. X, y, and other formal notations come later.
##### jsa09
not rated yet Aug 16, 2010
@cyberguy
@gunslingor1
Obviously the ( ) is supposed to represent an empty space where a number is to go. This is used in the very early stages before a child learns about x or y as variables - about the age of 6-8. In textboxs it is sometimes written as an empty square or an underscore instead of brackets. It is not a "notation" as such, just a teaching stage where the child tries to fill in the gaps. X, y, and other formal notations come later.

Why?

Let us start on the correct foot and save a lot of trouble later. I got sick of Primary school teachers simplifying stuff to such a level that it did not make sense at all.

How are kids going to learn maths properly anyway? "()" means something so why use incorrect notation and then have to relearn correct notation later?

Solving a problem with a single unknown is something any 5 year old can do and they don't care if you call it "X".
##### Nik_2213
not rated yet Aug 16, 2010
The problem here is confusing calculators' data-entry format with algebraic equations. I, too, was taught that the '=' derived from levelled scales, back when calculators were mechanical and the size of one-arm bandits...

Then you have the issues of 'infix' and 'post-fix' notation, which are not necessarily self-evident. The priority of +-*/() can also confuse. Context is all, and may be ambiguous...

Too often, these concepts are badly taught, as 'doing' rather than 'understanding': The latter may take several different approaches and many more examples before a concept 'clicks', may require individual tuition to find an approach that a struggling student grasps...
##### gunslingor1
5 / 5 (1) Aug 16, 2010
Cyberguy, I suspect you're a teacher with a teaching degree.

I think your ineffectively overengineering your teaching methods. You say "Obviously the ( ) is supposed to represent an empty space". How is that obvious? There is no instruction indicating it and these are not standard mathematical operations; I personnaly would have thought that ()=0 and would have given "not equal" as the answer. "Find ()" would have covered it, afterall it doesn't matter what you call a variable, you could even call a variable "~&^#%~poopy". That would have been fine. But, I've seen in these courses, you often just present the problem (especially on tests) without explaining it. The result is a guy like me who can ace every test without studying, skipping classes because I already know the math, but still failing the test because your teaching in a very confusing manner.

You need to instill the proper mathematical techniques and principles from the begging, especially "listing all assumptions".