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US Students Struggle With Understanding of the 'Equal' Sign
bickerd--- writes with news of research out of Texas A&M which found that roughly 70% of middle grades students in the US don't fully understand what the 'equal' sign means. Quoting:
"'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.'"
So I'm not being a curmudgeonly old jackass when I think this generation is stupid.
This is a sig. It is like every other sig in the world, except that it is mine, and it is different.
Well, no one was born knowing what the equals sign represents. In fact, it's been around only for 500 years. My personal opinion is that until we start forcing graduates of US Education programs to take at least a little math beyond passing out of algebra, the cycle is doomed to repeat.
FTFA, '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."'
Uh huh, see point 1 = 1 + 0 above.
This is one reason why we home school...public school systems fail in so many ways.
I blame it on calculators where the evaluate button has "=" on it.
One cause of the problem might be the textbooks, the research shows.
Which sounds a lot like the true cause, not the students - who in my case has an honours degree in physics.
politicians are like babies' nappies: they should both be changed regularly and for the same reasons
It means that even after China abolishes it's sweatshops there will still be a source of cheap unskilled labor in the world.
I have college diplomas in the fields of mechanical and electronic engineering (technologist and technician for the Canadians). I also took all advanced math, physics and chemistry classes in high school. I don't remember ever seeing the notation "4+3+2=( )+2" before.
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"'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.'"
4+3+2 is not equal to 9+2.
No kidding!!! What do you say at this point?
Kind of baffled to see "( )" instead of say.. x? I have never seen parentheses used like that, at least not that I can remember. In what region/mathematical area is this commonplace? You would think an article discussing not understanding basic symbols would actually attempt to use the most commonly used symbols in the argument..
And how about Economics, Politics, Aeronautics, and Quantum Mechanics?
To put it another way, he's saying that the students are treating mathematical expressions as a list of instructions to be obeyed, and not as expressions. This works fine for 1+2=? or 4/3=?, but leads to a cognative train wreck when trying to deal with even the simplest algebra. A student who works that way could never figure out what length of crossbeam they'd need to brace a 3x4 wooden frame.
No kidding!!! What do you say at this point?
I have a hard time believing algebra students would do something similar if you replaced the parenthesis with a single character (like an x) in 4+3+2=( )+2. I am not surprised that students are confused when presented with equations using unfamiliar symbols rather than conventional single character variables. I am also not surprised that pre-algebra math students don't understand algebra. Judging from the summary it looks like this research was setup with the specific intent to prove their preformulated conclusion.
It should be: 4+3+2=x+2 (Solve for x) I don't see the point in substituting parenthesis for a variable. It just makes it more confusing for everyone.
Thats what I gathered too, and it was a bit confusing to read. Knowing parenthesis as delimiters for so long, it was strange to see. I wonder if that is what they showed to the kids, and how it would have been different if they used something like:
4 + 3 + 2 = ? + 2
I think (4+3) is the best answer.
Me too, because I homeschooled.
Then I had Microsoft Office in junior high and Java in high school... what bullshit. Did programming suddenly get more complex, and now we can't teach it?
So that explains the MS Office Ribbon?
My first Journal Entry ever, in 8 years! http://slashdot.org/journal/365947/aphelion-scifi-fantasy-horror-poetry-webzine
"Researchers at Texas A&M struggle with Meaning of Parenthesis."
I am very small, utmostly microscopic.
That's not what = means. = is ASSIGNMENT. They're looking for ==.
Much as I know you're joking, I'd really love to get rid of this bane that C has brought upon us. Many previous languages used := to mean assignment, hence avoiding the clash with the mathematically well defined = symbol.
Come on, you can do better than that.
First you would need to prove a left-leaning portion of US society exists.
And then explain why the "education is for he who has the money and power" right wingers should be against their evil plans to make poor people stupid.
It sounds exactly like what right wing governments all over the world have been doing since there are governments.
Left wingers actually try to make people more intelligent through public education. Their problem is that their definition of intelligent is brain-washed.
It's mathematics or math or math's. It's really dumb to remove the ematic and leave that trailing s. More so when you leave out the apostrophe which one is supposed to use when one leaves out letters. Plus it's much more in keeping with general rules for pronunciation of English words. Maths is just awkward.
The problem is students memorize procedures without fully understanding the mathematics
That sums it up quite nicely. Students learn one way of solving a problem and memorize how to crunch the numbers to get the expected answer. This always bugged me when I was in school too. As soon as something didn't fit in nicely with what they had already learned, they'd be clueless because they don't understand what each value represents or why values relate to each other in a certain way. They're not taught to think for themselves. I rarely ever did homework, but I had a good fundamental understanding of the concepts that were being taught, so I "learned" more and never once worried about staying up late to cram for a test. This applies to just about every school subject, but is most obvious in math.
___, this is very ____ to do.
Where are you from, so that I can make up a ridiculous name for your nationality?
Gamingmuseum.com: Give your 3D accelerator a rest.
In the elementary and middle-school texts standard notation is rarely used. I've got a doctorate, but helping my kids through their math often is a real stumper. It is very common to use a box, a blank, or a parenthesis to indicate something that they are to fill in in a "number sentence". The theory seems to be that you don't need to teach about unknowns and variables because that would be confusing. So this notation is somehow intuitively obvious to the least observant. As they may not cognitively be ready for the concept it becomes even more obscure. Have a look at the books sometime - you'll want to scream. I can testify that the methods used up until the mid 1960's were MUCH more effective in creating mathematical literacy. The Stanford Studies Mathematical Group (SMSG) series of math texts was, to my memory, the flying wedge of what was termed then "The New Math". The strategies like 4+3+2=()+2 come from that movement. Truth is, the "New Math" is a dismal failure and resulted in the destruction of the mathematical competency of two generations of American students. Unfortunately the math teachers now all came up through that system and have no idea that there is a better way to teach math.
Actually, people should be identified by their state--Texan, New Yorker, Floridian, etc. It's not the United State of America, it's the United States of America--indicating that each one has a level of sovereignty, and people should be identified by that smaller area. Similarly, people are Scottish or Welsh, and not United Kingdomian.
Battlemaster--Game with friends in medival realms
Ok I'm going to display my ignorance here and ask why isn't = on its own good enough for a comparison?
.
I used various forms of BASIC as a kid, and = was fine there. I had some formal education in Pascal, = was fine there.
Now when I occasionally do a little scripting in a modern language, I spend most of my time tearing my hair out at bugs which turn out to be the result of me using = when I should have put ==
I'm sure there are good reasons for it that make sense to proper programmers, but personally I'd like to give whoever came up with this syntax a kick in the bollocks. Why would I want to do an assignment in an if statement or a loop condition check anyway?
This sig all sigs devours
If you have a ?, how do you write in the answer? Underneath the ? or above it or squeeze in the side?
Text books often use __, squares, and ( ) so people have a visual clue that something belongs there, before the concept of algebra sinks in.
Of course the true nerd knows that the operator used for this depends on the language. C and C derived languages (and thanks to the pervasiveness of C, most newer languages) use == for equality and = for assignment. But not all do so. Pascal for example uses = for equality and := for assignment, and so does Ada. BASIC uses = both for equality and assignment.
The Tao of math: The numbers you can count are not the real numbers.
That would be fucking awful to type compared to = and ==, though. Glad they didn't do that, it'd be so dumb.
The right: Education is for those with money/power
The left: Non-elites should all get the same level of education
The people: Everyone should be educated but not restrained
Yes, everyone should have the same opportunities for education. However, by lumping everyone together into the same education basket you implicitly restrain those who are capable of much more.
I am disallusioned with both parties because they BOTH support a notion of eliteness. I am more disallusioned with the left because, being a non-elite, they will lump me in with the rest of the crowd destined to become greeters at Walmart. At least the right lets me climb to the top of the non-elite crowd.
I was doing programming in the 4th grade, and this was in the early 80s...
I went to a rural public school in the 80s and I learned BASIC and LOGO starting in about 4th grade. In high school, in the time between when the AP tests were and the end of the school year, we learned programming, too.
I think they still break out the Lego LOGO with the younger kids, but by the time they get into the upper elementary now, if it isn't on the standardized test, they don't bother. This is a major factor in why I'm not a teacher.
There's a perfect xkcd for my sig but I'm too lazy to look it up. sudo someone go find it.
Math sounds awkward to me, because I was brought up with Maths. This is like an essay I read ages ago about why rear wheel drive is more natural than front. I thought it was a load of crap because I'd learned to drive in FWD vehicles and my natural driving instincts in certain situations were different to what the guy said that the "natural" was.
For most things in life whatever is more "natural" for you often depends on what you were brought up with/trained on.
which is totally what she said
Are you drawing that conclusion because you are a child, or because you've worked closely teaching math to young children for years? Or are you drawing that conclusion because you have a preconception about what the answer should be?
No one in my second grade class had an issue with algebra. We couldn't do division yet, but algebra was easy. The problem is not the students.
When you're afraid to download music illegally in your own home, then the terrorists have won!
Kids nowadays have ready access to technology, and are not adequately guided in its use. You can get a calculator in a dollar shop to do your arithmetic homework.
On a calculator, what does the = mean? It means "evaluate now". So that is perhaps where the running equals comes from. It is not a misconception. The students have correctly learned "evaluate now" from their electronic buddies.
The educators are just too obtuse to identify the source.
Let's take the example from the article:
4 + 3 + 2 = (calculator produces 9)
+ 2 = (calculator produces 11)
See? If you literally put in the symbols from the homework question into a calculator, that's what you get.
Now you might be able to ban calculators from the classroom, but the kids will use them at home.
Teachers should embrace calculators and explain how the [=] button has a different meaning which means "please calculate now", whereas the = used in math is a sentence which says "the left side is the same as the right side".
Yes, and no doubt the teacher worked a few problems on the board so that everyone could see how they were done.
And since everyone daydreamed through the class, the homework got done with the calculator.
4
+
3
+ (calculator displays 7)
2
= (calculator displays 9; write it down)
+
2
Now to finish the problem? Well, = of course, and write down “= 11”. That’s what the calculator said.
Alexander Peter Kristopeit bought his basement from his mommy for one dollar.
This is what we get when we have a society that values the celebrity and athlete more highly than anything else. This is what we get when parents think socializing is more important than good academics. And ultimately a lot of the blame falls on the teachers as well, for not doing their job properly.
Americans seem to think throwing money at our schools will fix everything. They also seem obsessed with small class sizes. That's something I've always found utterly ridiculous considering in Asia you'll routinely find classes with 30+ students and they are better educated than American students in a class half the size. Too much of our educational system has gotten too obsessed catering to the slowest kid in the class and making things fun. So instead of trying to bring the slow kids up to speed we're instead slowing the rest of the class down.
True enough about BASIC. However one of the worst design flaws in C is the combination of using = as the assignment operator together with the liberal interpretation of what constitutes an expression. How many lifetimes have cumulatively been wasted because some tired programmer wrote "if (x = y) ..." and the compiler raised no objection? Let's be honest, C is the king of side-effects.
In a sane language, = would not be used as an operator at all, neither for assignment nor equality test. Neither is what the symbol means in a mathematical equation, and allowing it for either is asking for trouble.
BASIC is perfectly sane. There are clean, contextual rules which disambiguate between = the assignment operator and = the equality test.
Let's take a moment to remember that "x = 1" is only a legal BASIC statement in the first place because interpreters have been relaxed for programmers too lazy to use "Let".
DRM: Terminator crops for your mind!
Perhaps kids should be taught to use RPN calculators.
On an RPN calculator, the keys which perform operations are labeled with symbols that represent mathematical operations. There's no misuse of '=' to mean 'perform calculation'.
GCHQ Quantum Insert installed. If only our tongues were made of glass, how much more careful we would be when we speak
but...but...I'm not an Indianaian, I'm a Hoosier!
Back then (80s) programming was the only way to use the TRS-80s and Apple IIs the schools gave us. Today? You just need to learn how to turn them on and click an icon, and so programming is no longer considered necessary unless you're going into a CSE major.
BTW:
I see a problem with the problem in the summary: 4+3+2=( )+2 is not the way math questions are typically phrased. In my experiences these problems usually looked like this: "4+3+2 = __+2 ; Fill in the blank." The instructions were explicit so students did not need to guess the teacher's desired result.
I don't like teachers that think writing confusing tests (aka trick questions) is any test of student ability. It's more a demonstration of the test-writer's lack of communication skills.
"I disapprove of what you say, but I will defend to the death your right to say it." - historian Evelyn Beatrice Hall
The students are the one who made up the =11 part. Try punching it the question "4+3+2= +2" into a calculator and you'll see why. To the students raised on calculators, "equals" doesn't mean equality anymore, it means "what do the numbers up to here add up to?" So they get to " = ( ) " and perform the "what do the numbers up to here add up to" operation, and write the answer in the blank provided. Then they're left with the +2 bit, so they add it again.
Left to right order of operations, for all operations.
If I have been able to see further than others, it is because I bought a pair of binoculars.
You know, that's exactly the kind of thing used to deny blacks the right to vote after the Civil War.
I'm out of my mind right now, but feel free to leave a message.....
The problem here is not the use of the equal sign, it is their completely asstarded implementation of the parenthesis that is some how intended to imply one variable twice, with a line break in the middle.
The parenthesis weren’t what triggered that interpretation; the equals sign was. Exactly like a calculator: you calculate, you push “equals”, you get an answer. You calculate some more, you get a new answer.
Alexander Peter Kristopeit bought his basement from his mommy for one dollar.
Someone already said: it is a operator precedence problem, not about students' interpretation of the equal sign. Looks like the researchers could not pinpoint where the misunderstanding is.
It's too bad I already posted in this thread and can't mod this up. You're precisely right. The government education system abandoned learning a long time ago. Today it far more akin to another government make-work welfare program.
Granted, there is something to be said for a well-designed, visually appealing text book, as long as it has actual material in it. I have some of my Dad's old Schaum's outlines from the 60's and 70's, and it's damn near impossible to learn anything from them because of the sheer density of material.
The problem is that
America is a continent
Except... It isn't. At least not on any globe I've ever seen. Not anymore than "Dakota" or "Carolina" is a state.
Of course, you Europeans love playing games with the names of large land masses to further your own ends. It was you racist honky bastards that decided "Europe" was a continent instead of what it actually is, a part of Asia. But, you couldn't bear your pure white homeland to be infringed by the dirty dark skinned peoples could you you racist piece of detritus? You cracker motherfuckers have something coming and when Islam takes over, we're going to be coming to give it to you.
Captcha: tribute
Imagine that.
___, this is very ____ to do.
I can find at least two solutions :
This used to be the norm, but I think our civil war put pat to that idea as the nation become more important than the individual states. Even then, which one would it be? My state of birth? The one I grew up and was socialized in. And if more than one, which? The current state of residence? The last one I paid taxes in? I'd bet most people move from state to state at least once in their life and often more due to schooling and work. I suppose mine would be Washingtonian as that's where I live, although I prefer the term Okie as I grew up in Oklahoma and left never to return (where people who still live there are Oklahomans).
Maths is awkward? Math's is just plain bonkers. How did you arrive at that abomination?
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If you teach and expect kids to be stupid, they will be.
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Actually I observed more people in CS/college to have more left leaning socially supportive views than any other class.
By contrast, the poorly-educated-blue-collar work force, I've found more Rush Limbaugh listening imbeciles with completely socially deviant views, absolutely ignorant of the reality about them
To me, this clearly indicates how we can have a government for 10 years that ass rapes it's citizens, especially their low income supporter base (Rush Limbaugh & Fox news viewers), whereas when a change in command comes in and wants to support the less privileged, these same people dig their nails in for a fight to the death, instead of change for a better life and opportunities.
Completely mind boggling. America is reviving the dark ages that Europe went through during the middle ages. Lead by the conservative christian right and religious superstitions of men.
"Suppose you were an idiot...and suppose you were a member of Congress...but I repeat myself." Mark Twain
...and I had a very hard time understanding why one would put anything other than a 7 inside the parentheses.
Then it dawned on me that, apparently, some US students interpret the "equals" sign as a "write the result of the preceding arithmetic operations" sign, which the students promptly do. Then, they see the "+2" following the parentheses, and are completely dumbfounded by it, so they assume there is a missing "write the result of the preceding arithmetic operations" sign, which they add, so that they can enter the result of "9+2" after it. Presumably, "+" does not mean just "plus", but “add these numbers and write the result after the "write the result of the preceding arithmetic operations" sign”.
Wow!
That all depends on the school, and the classes the school assigns a student to.
I was introduced to programming in the early 1980's. The school bought a TRS-80 Model III, and it was given to the gifted class. That was when I was in primary school. No one had a clue of what to do with it. I got my hands on some programming books (the good ol' printouts of basic programs) and started learning. We didn't have any software to run, so that was the limit of what we could do, and most people had no interest in it at all. Heck, most kids couldn't even type then.
Later on, still in the early days of computers for students, weren't taught how to write programs, we were simply instructed on how to run programs. "insert disk, type this, follow the prompts". As we started getting computers at home too, some of us started programming.
I would strongly suspect that it is different now, but I could be mistaken.
I think the article is misunderstanding the confusion. Children are being taught "1 + 1 = ". the equals sign means that they take the formula on the left, and calculate it to put on the right. It would seem to be a logical extension of that to use the equals sign to indicate a calculation should be done, not that both sides are equal. It's not a problem with their ability, it's that the idea hasn't been explained to them. It wasn't until I was in Algebra that the idea that the equals sign really showed that both sides were to be equal, and that you should solve the problem accordingly.
Serious? Seriousness is well above my pay grade.