COLLEGE STATION, 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 math 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 students. 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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About research at Texas A&M University: As one of the world’s leading research institutions, Texas A&M is in the vanguard in making significant contributions to the storehouse of knowledge, including that of science and technology. Research conducted at Texas A&M represents an annual investment of more than $582 million, which ranks third nationally for universities without a medical school, and underwrites approximately 3,500 sponsored projects. That research creates new knowledge that provides basic, fundamental and applied contributions resulting in many cases in economic benefits to the state, nation and world.
Contact: Keith Randall, News & Information Services, at (979) 845-4644; or Robert M. Capraro, Department of Teaching, Learning, and Culture, at (979) 845-8007; or Miao Jingang, News & Information Services.
Tags: Equal sign; math education; student performance, Robert M. Capraro
When you presented the problem ” 4+3+2=( )+2″. What did you tell them to solve for?
If you had stated: [4 + 3 + 2 = x + 2. Solve for x.] It would be pretty hard to believe many people could not do that.
However, if you are just putting out there “4 + 3 + 2 = ( ) + 2″. What the heck do you expect someone to do? Guess that you want them to solve for the whitespace you didn’t signify was a variable?
Perhaps if the teacher used “x” instead of “()” as per the standards for basic algebra there wouldn’t be so much confusion.
What is this () nonsense, inventing a new way to show a formula and then claiming confusion is a result of stupidity is misleading. I doubt if they had shown it properly, as 4+3+2=x+2, that anyone would have struggled.
The confusion is with the nonstandard equation format. 4+3+2=( )+2 should read 4+3+2=x+2, solve for x. There may be deficiencies in US schooling, but this study has no credibility if it uses such fundamentally flawed methods.
I think this research shows that if you use non-standard symbols or terminology the subjects might get confused. There is a standard way of presenting the problem you wanted the subjects to solve 4+3+2=x+2. If it had been presented that way almost everyone would have answered correctly. I have never seen this use of symbols: 4+3+2=( )+2. To me that is just undefined.
If you set students physics problems in French and they got them wrong, what would that prove?
I find your answer to this equation 4+3+2=( )+2 insane. To me this equation says the answer to the left is equal to the answer on the right. So in the parenthesis the number will be 7. I have seen many examples like this when I was younger and once again mathematicians have gotten it wrong. You take inferences that I know what you want from this equation without providing more information.
Sad!!!
Ram
This is actually quite flawed, because the notation, 4+3+2=( )+2 is not the “standard” way of writing a formula (assuming that is what it is.) It would normally be written as 4+3+2=x+2 where one then solves for ‘x’. When would this notation actually be used? If it is some new teaching method, then it’s a very, very risky method, because it is too confusingly close to standard formula notation. I’d hate to be on an airplane expecting to land safely on a runway knowing that the formulae used to get me there safely may or may not be correct.
So I don’t think it’s really the students “memorizing symbols” that’s the issue, but the teachers who are teaching methods that ultimately can be confused with more standard methodologies.
Very interesting point.
I think one of the problems is that, at least in Texas, we use the word ‘Equals’, and the word is used as a verb.
“4 + 4 equals… ”
This imparts a sense of ‘becoming’, as though the ‘4 + 4′ part isn’t something until it equals something else.
In other languages, notably Chinese, the word used in math (so I’m told) is equivalent to the English word ‘is’, conveying the concept that one side is equivalent to the other.
Parenthesis usually do not denote a variable, maybe you should fix your process first.
Why are you using “4+3+2=( )+2″ instead of “4+3+2=x+2″. I might have been out of the loop from school in a while, but this is the first time that I’m seeing a pair of parenthesis replacing a variable in standard algebraic notation.
More like teachers have a limited understanding of how to symbolize mathmatics. () is used to denote 0. Yes 0 so, you would be stuck here with an inequality.
4+3+2=()+2
9=0+2
9=2
WRONG… Horribly wrong. Nice journalism there buddy.
Shouldn’t there be a variable??? We used to solve for x, not solve for (). No wonder they are confused.
Or, perhaps, Capraro screwed up by not using basic standard algebraic notation, in which the problem would read 4 + 3 + 2 = x + 2. In that form it is obvious what the “question” is. “4 + 3 +2 = ( ) + 2″ is vague as to what operation the asker wishes to be performed because ( ) has no standard meaning.
…or possibly the confusion could come from using parenthesis instead of a variable like “x”. Teacher Fail.
Duly note that this article doesn’t explain what the equals sign means, either.
Students who have not learned what ( ) means (in *this* example) will not know what to do with that mess. Why should they? That wasn’t in my math curriculum. Using “n” would have been simpler. Was that avoided on the grounds that students shouldn’t be expected to understand variables yet? If so, then why on earth present exactly the same conceptual demand to them, but in less standard notation?
Here’s my challenge: present this little blurb to 100 teachers and ask them to summarize it.
(I’m also curious as to how comprehension of the meaning of the equals sign was determined here and in other countries, and whether students learning under one curriculum in this one fare better than those at the mercy of another curriculum. *Do* those students doing “Everyday Math” end up understanding better? OR do they simply (as I’ve seen, but in very small samples) memorize different procedures?
Headline should read “Researchers at Texas A&M struggle with Meaning of Parenthesis.”
Cultural bias. I’m American and I’ve never seen parenthesis used for a variable. In classes prior to algebra, I would have seen this problem written thus: 4+3+2=_+2 .
This is a very poor study if the problem was presented to students the way it is presented in the article. The notation of using parenthesis to represent the variable might confuse many college students since parenthesis are properly used as delimiters in mathematics. The use of a variable such as x or even a question mark or blank would very likely have returned very different results. I don’t believe the conclusions drawn from this study are at all “equal” to reality.
4+3+2=( )+2 depends on how the question is being asked of the student. If you don’t want them to “Run the equal sign.” don’t ask the question in that manner.
A different way would be to ask*:
Please place parenthesis around the numbers that makes this correct and find the missing value.
4+3+2= ? +2
(4+3) + 2 = 7 + 2
9 = 9
* This is the way the Everyday Math program out of University of Chicago (http://everydaymath.uchicago.edu/) works.
Notice that this continues the illustration of how equality is used all the way down? A subtle point many many teachers forget to do. Something I was taught 30+ years ago that somehow became lost. I started to show my 3rd grade son this method and now he ‘gets’ it.
This method easily translates into the pre-algebra system of 4+3+2 = x +2 and can be expanded to 4 + 3 x 2 = 16 – ?
4 + 3 x 2 = 16 – 2
(4 + 3) x 2 = 14
7 x 2 = 14
14 = 14
In short, you have to ask the question in a manner that makes sense. Many times the question is ambiguous or a properly developed example is not provided to aid in clarity.
Last point – you need to have a teacher that is solid in math teaching math – at all levels K-12. K-4 is the most critical. While true for all subjects, math is one of the few subjects that actually introduce completely new material in later years. Language arts you have prose and poetry. That’s about it. Everything else is refinement.
Too many students go into elementary ed. because ‘they didn’t like math’. That student should automatically be excluded from teaching until they get both an attitude adjustment as well as proper training on where they are deficient. A teacher that doesn’t like a subject will show that to their students. Who will then no like it either.
Fred in IT
Why the use of “4+3+2=( )+2″ and not something more standard like “4+3+2=x+2 Solve for x.”? Why use parenthesis when parenthesis normally indicates that “(this)” part of the equation is to be evaluated first but yet there isn’t any contents there? Of course kids are having a hard time understanding this equation.
How about sticking to the good old basics of Algebra? Instead of this ( ) nonesense, the example above should be: 4+3+2=X+2 And the student should know to “solve X”. It amazes me how we change things that were not broken and then wonder why it’s now broken.
I’m sorry I am forced to quote a comment from a different discussion, however I share the commenter’s sentiments:
“Since when is a set of parentheses a proper substitution for a variable? Seriously, part of the problem is that the standards for writing and evaluating mathematics in (especially) earlier grades is subject to what’ I’ll call “local interpretation”.
As the father of a rising third grader, and a professional engineer with masters degree that included more math than I care to admit, I’ve puzzled over the way problems are written. At least one in ten homework assignments require that I look at the answer sheet to determine what the question is actually asking. Some of the answers appear to be wrong, except when interpreted in a very specific way which is counter to standard practice. Others are simply misleading.”
In other words – dear lord, what kind of kooky notations and explanations are used in the US education system because of this well-intentioned, yet horribly-executed quest to make math more understandable?
Going through school in New England I remember being taught that the equal sign represented equality on both sides and not just another operator. It would be interesting to see the distribution of understanding within the US alone. Did one part of the US rank higher in understand than others?
The next step is to look for root causes and determine the one most likely to be the cause of this. I would like to suppose the root cause is the attack from the religious right on the school curriculum, you know, show then evolution and ID and let them decide mentality? Or show them the equal sign and let the students decide what it means, so they feel good about themselves? I know that’s facetious and jumping to cause. So some real research and repair is in order. This issue IMHO far exceeds the global warming religion.
I found this extremely interesting. I’ve noticed something like this from tutoring a few high school age students. In calculating say sin(x^2)-x^2 when x = 0.5 to 3 significant figures it’s common to write
sin(x^2)
= sin(0.25)
= 0.2474039-x^2
= 0.2474039-0.25
= -0.00260
It strikes me that this is the same logic used in long division and multiplication – write out the first step, and then use that to do the second step, and so on. I get the impression that students understand in theory what an equal sign means, but that these methods have made them used to doing running calculations. Would this be an example of the mistake that was identified here?
Thanks again for posting this here,
Simon
If the students were actually asked to solve “4+3+2=( )+2″, then I understand how they would be confused. This notation is non-standard and does not mean anything at all. If on the other hand it was not what they were asked, and as the video suggests they were asked to solve “4+3+2= __ +2″, then please correct your article.
One of the most useful things I ever realized about the equal sign was that it meant “the same as”. So 1+2 = 3 could be read as “1 plus 2 is the same as 3″. Once I realized that, it was very easy to start making the move into understanding algebra. I think we depend too much on memorization in American teaching. Understanding is the key. Without it, we’re nothing but a bunch of parrots.
4+3+2=(9)+2=11 is correct when using a calculator. = means evaluate in that context.
Where did you come up with the () notation?
Nearly every other algebra problem would have stated it as 4+3+2= x+2 solve for x.
The () implies computer code where the = symbol is used an the assignment operator.
I don’t like you example: 4+3+2=( )+2
It’s not clear to me exactly what you want… but it’s not the meaning of the equal sigh as much as the meaning of ( ) that confused me.
If you had said 4+3+2=?+2 I believe it would have been more clear… the distinction being a question of what should go here versus a blank to be filled in… the original lacks some clarity in my mind…. so I would be tempted to solve the left side and having it’s value to put it in the blank and then to solve the right side, not seeing the equation as a question to be answered as much as an equation to be processed… I know as clear as mud.
If you have not done so, consider checking on how you represent the variable to be solves, back in my school days we used ? before algebra and then variables such as X.
You should read my rant about the equal sign here.
http://lifecs.likai.org/2010/08/equal-sign.html
Good luck achieving racial equality when the concept of mathematical equality is so elusive. #onlyintexas
This is just stunning how kids are being taught to grow up stupid. I hope this discovery about it, changes the US school system to ensure the kids learn to understand the equal sign.
This is middle grades math so the basics of algebra won’t be introduced to these students for a few years. And the point is not that they use a blank for something unknown, it’s that the equal sign is misunderstood.
To my calculus students the equal sign often means “and the next thing I do is…” For instance “lim_{x->∞} 1/x = 1/0 = DNE”. How many things are wrong here? 1/0 is not a number so it can’t be equal to anything. DNE is also not a number so it can’t be equal to anything. And “…equals ‘does not exist’” is a predicate with two verbs.
I’m a physics graduate student that got a minor in math. I’m pretty sure I’ve never seen 4+3+2 = ( ) +2 before. Maybe more kids would get it right if you wrote 4+3+2 = x+2. At least they would question what the x means. What is shown could easily be interpreted as the “mistaken” instance, since it isn’t really defined. Were the kids told what it meant?
If I give students a basic problem like 3x+2 = 17, several of them are going to write an answer like 3x+2=17=3x=15=x=5 and I just know that those are the students who are not going to get non-basic problems.