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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.'"
That's not what = means. = is ASSIGNMENT. They're looking for ==.
Also, on a serious note, from what I recall of the US school system, frankly, the most surprising thing about this is that the problem isn't worse than reported.
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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.
Maybe the students figured they would get extra credit for going the extra step?
This is a bit weird. I mean how bad is the lack of understanding? Bit hard to follow the article to be honest, is it just because of the 'everything in one line equal to each other' ? Or does it also include more complicated stuff like "1 = Number" ?
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.
I am terrible at math; I had to take the most basic math course twice in college. However I can not begin to comprehend how fucking stupid you have to be to not be able to properly answer 4+3+2=( )+2. I guess they can always become philosophy majors like I did.
Don't blame me, I voted for Cthulhu.
Schools in america do not reward understanding of the mathmatics behind the symbol.
They reward memorization.
Don't blame the players when the game is retarded...
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
Didn't they just fool the students with odd / non-standard use of symbols?
I presume that 4+3+2=( )+2 is supposed to mean the same as 4+3+2=x+2.
If they had presented the equation with x, surely (almost) everyone would have solved it?
I'm from the UK, is 4+3+2=( )+2 a commonly used / commonly understood way of presenting the problem in the US?
I think the U.S. math curriculum could use some equality with the Chinese system.
sheeeeshhh
Because I can't figure out how you are supposed to solve such a problem, and I have a BS in Computer Science.
Let's look at the problem:
4+3+2=( )+2
4+3+2 = 9
( ) + 2 = 2
So we have a false equality 9 = 2
Since this is not true, I can easily see how lots of kids would go through contortions to try and make it true.
But unless this is a trick question, why are the setting up false equalities like this for grade school kids?
A work that expires before its copyright never enters the public domain and thus enjoys eternal copyright protection.
It means that even after China abolishes it's sweatshops there will still be a source of cheap unskilled labor in the world.
Let's see here.. I'm going to go with:
4+3+2=(21/3*981727612785316256514034236^0)+2
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.
UNIX/Linux Consulting
"'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?
I mean, I guess I just never thought of it that way.
I was aware that people solve math problems differently, mostly from discussing methods of figuring the tip at restaurants (I just round off to the nearest 5 and divide by 5 to get about 20%) however I never considered that someone might not learn the meaning of the symbols that they use.
I would like to know more because, I understand the equals sign, but I still use that a + b = 1 + 2 = 3 notation when I am just calculating something for a quick note, since I don't care about formality, I just want to have the result and the values used to calculate it so that I can check my work later if need be.
-Steve
"I opened my eyes, and everything went dark again"
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..
Obviously the US students are now totally confused about what equality means, everything is equal to everything else, your work effort is equal to anybody else's work effort because based on liberal agenda the outcomes are supposed to be equal.
So clearly, women=men, black=white, all humans have equal rights, this inevitably leads to everything else being equal to everything else.
so 11=9, 0=1, Islam=Terrorism, America=Fuck Yeah=One Nation Under God=Obama=God Bless America=And No Place Else=Nuke The Whales=There Is No God=Gay Is Good=Gay Is Bad=Government Is Going To Fix Everything=Large Corporations That Are Monopolies Because Government Made Them Monopolies Are Going To Fix Everything
So you see, these students maybe confused, or maybe they are right and everybody else is stupid for not getting with the times.
Obviously.
You can't handle the truth.
Sure, inability to understand basic arithmetic leaves students unprepared for work in science... or engineering... or operating a cash register... or keeping society from crashing and leaving behind a postapocalyptic wasteland. But *in* that postapocalyptic wasteland, the cannibal hordes will find innumerates to be just as delicious as anyone else, so learning math would have been a waste of time anyway!
Wow. I can't read. Heh. Thanks!
When you're afraid to download music illegally in your own home, then the terrorists have won!
In my (very) small town school in the 70's and 80's, we didn't really learn algebra until 9th grade. Up to that point, the equals sign was pretty much used only as the "answer symbol", meaning "here is the result of all the stuff on the left".
Strange things are afoot at the Circle-K.
I note that the example cited in the lead article, is simply what you would get if you hit those keys on a calculator.
If only we all used RPN, and there was no equals sign!
I am also extremely frustrated that all calculators have a divide button, but very few have the "gozinta" button. "Gozinta" is a way more useful concept than "divide" most of the time.
e.g. "7 gozinta 42" has the answer "6 times". I can do that rather painlessly on most RPM calcs with xy but on most other calculators it's more like "7, I already typed that in, I'll hit 1/x, then multiply by 42 and hit equals".
And NOW I finally understand it. To add an example of my own [1+2+3+4+5]
1 + 2
= 3 + 3
= 6 + 4
= 10 + 5
= 15
I blame overuse of calculators if that's the case.
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.
That's what I was trying to understand. Why not use the traditional "x" for the unknown instead of the non-tradition open and closing parentheses "()"?
This doesn't show that kids do not understand the equals sign.
This seems to show that kids do not understand the what they are supposed to solve for in that example. They do not understand the meaning of empty parentheses.
And frankly, I wouldn't be sure that I had solved it THE WAY THE PERSON WHO WROTE IT THOUGHT IT SHOULD BE SOLVED if I had just substituted x for () and gone from there.
The more I think about this topic, the more I see the fault as being in the way the problem is presented, not in any lack of understanding in the students.
politicians are like babies' nappies: they should both be changed regularly and for the same reasons
For an expanded explanation of what the equals sign means, check out Petkovsek, Wilf and Zeilberger's A=B. I remember it as a very enjoyable read from university, in parallel with Concrete Mathematics... (btw, why won't š show in comments?)
-- Free speech is only free if your time is worth nothing.
Ya, what's up with that? We were never allowed calculators in university, why do they give them to kids?
When you're afraid to download music illegally in your own home, then the terrorists have won!
I think (4+3) is the best answer.
Did they try to use something else besides parens? Growing up, to solve math problems, I had to fill in a square box, or place the answer above an underscore. Parens actually have meaning in mathematical notation, so perhaps that is the real source of confusion, and not the equal sign.
I've noticed and written about this previously. I don't even blame the students that much; I don't think I was ever explicitly told what the symbol meant either. In standard curriculum you either have to pick it up inductively or you're crippled. Quoting myself:
But here's another way of looking at it: Each line of math is, effectively, a sentence. (A highly condensed sentence in specialized notation, but the same nonetheless. It can be re-hydrated back into normal English at any time.) And the equals sign is the verb "to be". It's the most important verb in any language! What if someone were in a writing class and submitted a paper without any verbs? What if they were entirely unable to say "you are", "I am", "he is" anything at all? Would an English teacher totally flip out? You bet they would.
http://angrymath.blogspot.com/2010/03/equals-signs.html
We know where leadership by an anti-intellectual "strongman" who scapegoats minorities and likes boisterous rallies goes
Do your Barbie dolls say "Maths are hard"?
Great news! We've found our new test at the polls. Present them with this:
4+3+2=( )+2
And if they cannot correctly evaluate it, pitch their chad/digital vote into the burn bin.
It's a pretty obvious test. Something + something else = Mystery number + the same something else.
If you can't figure it out... well, you're probably better at memorizing facts than thinking.
Futurist Traditionalism
Until today I have never seen a problem in this form i.e. "4+3+2=()+2"
4+3+2=x+2 yes. But ()?
??
While it is a simple problem and easily solved it did cause a slight moment of mental pause to figure out what I was looking at.
Kind of hard to draw conclusions based on what amounts to a trick question.
I am very small, utmostly microscopic.
It's not the calculators... it's the students and teachers. You cannot blame a machine for students either failing to understand or just never grasping that going from "4+3+2=( )+2" to "4+3+2=( )+2=11" is nonsensical. Don't make excuses for them. I say this as someone who barely got through math classes (and being 27, I'm in the same generation as most of these kids), and even I looked at their thought process and muttered "W... T.... F....?"
So that explains the MS Office Ribbon?
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as others have mentioned, the "()" aren't being used right to begin with... they are usually used to hold another equation that should be calculated before continuing on with the rest of the equation... i.e. 6 + (12/3) = 10
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Thanks to the RIAA, I buy them used...
"Researchers at Texas A&M struggle with Meaning of Parenthesis."
I am very small, utmostly microscopic.
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.
5 = 4.999999....
WTF?
That's a question of order of operations, not what the equals sign means.
The student in that example was presented with a vague question in a form which I have never seen before and chose to interpret it as two different questions tied together. I'm not going to try to argue that the answer is "right", only that in the absence of an explanation of what "a big blank space in the middle of an equation" means it's an understandable misinterpretation.
And then, because this is /., I will throw in the obligatory XKCD and suggest that it's somehow like complaining that a student in drivers' ed who has only driven automatic transmissions will naturally have trouble behind the wheel of a real car.
Since rote arithmetic is now passé
'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 because students are taught with years of 1 + 2 = ( ) "Fill in the blank"
They are suddenly supposed to understand that "= ( )" doesn't mean "do the math on the left and put the answer on the right"?
There's a teacher fail in the lower grades. They should be teaching "=" as "equals", ie 1=1, 5+2=3+4, etc.
Although, I bet if they changed the problem in the article, students would get it quickly. 4+3+2=2+() There's no "= ()" to screw with their minds.
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.
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.
Is it just my observation, or are there way too many stupid people in the world?
That's why my daughter is in private school!
I also wonder how these kids cope when a second variable is introduced.
I think the idea is that calculators are used in the real world for real calculations (counting a computer as a really overgrown calculator), thus teaching kids without using them doesn't prepare them for the real world*, they don't actually NEED all that skill with doing basic math in their head, because most won't be doing it regularly enough for it to matter.
I don't entirely agree with this, I think that they need enough skill to be able to tell when there has been an error in their figuring.
I don't read AC A human right
We were never allowed calculators in university, why do they give them to kids?
Actually, there's an interesting insight in that statement. If most of the arithmetic you've ever done is just poking digits and operations into a calculator, this math problem looks like a simple chained addition with a running equals sign. I'm not sure if any other of the commentators in this thread picked up on this (mostly because you'd have to RTFA, and we know that's pretty much unheard of.) The summary doesn't mention calculators, but the entire cargo-cult handling of numbers implied by this reported mathematical literacy problem is pretty much driven by calculators.
Kinda makes me think we should abolish infix-entry calculators and stick with RPN like God intended.
Welcome to the Panopticon. Used to be a prison, now it's your home.
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.
That was my first thought, but their usage is even worse.
4+3+2=()+2=11
If we treat () as variable x, we get:
4+3+2=x+2=11
Simplified:
9=x+2=11
We now have an equality operator with two non-equal values.
What they are implying is:
2+3+4=x; x+2=11
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.
-Rick
"Most people in the U.S. wouldn't know they live in a tyrannical state if it walked up and grabbed their junk." - MyFirs
US Students Struggle With Understanding of the 'Equal' Sign
The title makes this post sound like a joke. Wouldn't a better title be "Some U.S. Students Mathematically Illiterate". Given that 14% of the U.S. population are practically illiterate, it comes as no surprise.
The whole thing is a misunderstanding of operator precedence, that's all.
Ask the same people about "8+3*4" and they'll probably get that wrong too.
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.
My wife is taking calculus right now at a local college. Her professor takes points off if she solves a problem using intuition or innovative thinking rather than following the procedure he outlines in class. She gets the right answers, but gets marked "wrong" for not doing it the way HE knows it works.
Basically, the professor doesn't understand the material himself, but rather knows the procedure for specific types of problems, so that's all he can grade on. The same problem is prevalent in secondary schools as well. Teachers do not understand the material they are teaching outside of a vary narrow scope of memorization, or a rigid state-run curriculum from which they are not allowed to deviate because of the risks involved with individual free thought (like lawsuits over speech or utterances that are not politically correct, or that hurt the childrens' feelings, and so on).
Texas school boards defend teaching alternative meaning of equals sign
'The students have the right to hear both sides in this debate. What is currently taught - that the equals sign means that both sides are equal and cannot mean "evaluate the left-hand side" - is biased under the influence of powerful left-leaning groups, calling themselves "scientists" and "engineers". These are the same atheists that say that creationism, as dictated by the bible, is not a valid alternative for their pet theory of evolution. Their "algebra" is purely based on politically and religiously biased axioms. Their definition of "equals" confuses students trying to understand things such as pocket calculators and the concepts of "gender equality" and "racial equality" in society.'
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.
Well, I would, of course, put in (the answer to life, the universe and everything)/6
The Tao of math: The numbers you can count are not the real numbers.
It could be the same study; I didn't RTFA; but I've read at least one study recently that was bemoaning the use of "=" as an instruction to "do something" (like "evaluate what you saw on the left"), rather than a statement of fact. The study goes on to say this causes the thought process of actually finding equalities to be less developed so trying to do basic algebra requires some RE-training.
I think the people latching on to the use of parenthesis are missing the point. Then again, maybe I am.
Before you design for reuse, make sure to design it for use.
There are some respected mathematicians that will tell you that every equation is a lie. The meaning of the equals sign depends on the context. For example \sum_{n=1}^{\infty}a_n=s means
that the limit of an associated sequence of partial sums is equal to s. This story doesn't even some close to revealing the problems with the "equals" sign.
The problem is that it is not stated directly what the question is that is being asked, so students are required to infer what the question is. Unfortunately, they just assume the "=" implies a question is "what is 4 + 3 + 2?" since that is the sort of question they are used to being asked. They are probably not used to (a) seeing an equality as a "statement", and (b) inferring from the hole in that statement that there is an implicit question as to what would fill in that hole to make it true.
It's a very easy leap for me, and probably always has been, but for my second-grade son on the autism spectrum it is a nearly insurmountable leap of logic. Our local school system works very hard to encourage this conceptual thinking in math, but at least for my son that is making it even harder for him to shore up the basics.
All that to say, I can understand how this state of affairs could come to be, but am sad to see that middle school kids in general are not exposed to enough logic and inference.
Hmmm ... I just noted that 6*9 / 6 is, of course, 9. So maybe the true problem is all of them knowing the Hitchhiker's Guide to the Galaxy, and the testers didn't get the joke :-)
The Tao of math: The numbers you can count are not the real numbers.
I believe := started with Algol.
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Calculators are allowed in universities now. Hell we even have sophisticated math solving programs. Mine was "Maple".
When I was studying multiplication, I just could not comprehend it. I was getting failing grades constantly, while my classmates were memorizing their multiplication tables and acing exams. I just couldn't understand it. Then, it dawned on me... multiplication wasn't some new mystery math, it was just addition in a new form!
Then I became better at multiplication than all my classmates, and stunned the teachers by how I went from getting 80% of my tests wrong, to getting 100% correct, and faster than my classmates. Unfortunately, it was at the tail end of the unit, so I still got a bad grade on my report card.
The teachers thought I was cheating, too. They had me take tests in front of them, during recess, to prove that I wasn't cheating. They then accused me of being lazy and not paying attention previously. No, I just didn't understand the mystery math they were trying to teach me, because they were expecting me to memorize things, and not actually teaching me to understand it. I don't think they ever accepted that truth.
So it is with addition and =. Children are taught to do this, then that. They are taught process, not meaning. They need to be taught from the bottom up, not from the top down. Teach them that = means equality, not evaluation.
Oh, and use standard notations, not this ( ) garbage that nobody uses.
Seems like a bad transcription of an interview with the researcher.
In the actual journal article they either used an underline or nothing at all, eg. "6 + 9= __ + 4" and "__ + 3 = 5 + 7 = "
I'm also not really surprised since as other posters have mentioned I don't really remember learning any real algebra until 6-8 grade. This really just means that Chinese kids learn this stuff earlier than US kids and maybe US kids should learn it early on as well.
Maybe they deliberately used non-standard notation to see if the students could come up with a sensible interpretation of the problem.
Yes, writing "4+3+2=( )+2" when you mean "solve 4+3+2=x+2" is weird, but interpreting "4+3+2=( )+2" as "compute 4+3+2, then add 2 to the result" (which the example of a wrong answer given in TFA) probably means you don't understand what an equal sign is.
The bible comes this close to giving us the square root of 2.
Rev 21:16 -- And the city lieth foursquare, and the length is as large as the breadth: and he measured the city with the reed, twelve thousand furlongs. The length and the breadth and the height of it are equal.
And this close to giving us pi.
2 Chron 4:2 Also he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about.
Oh well.
-fb Everything not expressly forbidden is now mandatory.
As a high school math tutor for one of the large national test prep companies, I can say with complete certainty that even when the "box" is replaced with a variable name, students still do this. I find that it's not so much that they don't grasp that the = symbol means equality as much as that they don't realize what equality means.
If I give a problem like:
f(x) = 2x + 7
g(x) = sqrt( f(x) )*5
Many students can solve this problem; a good portion of them just understand "oh, I do a substitution" - they don't understand fully that the equality of a variable with some expression is WHY their answer is correct. Similarly, something like: 3/15 = x/225 Most students get thsi right; "oh, I just cross-multiply" - but they don't understand WHY cross-multiplication is the correct thing to do. Teachers need to start teaching real math in addition to problem solving methods and heuristics.
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".
Depends how you interpret the '=' sign. If you interpret as 'next step', then:
4+3+2=()+2 add the numbers and put them where () is. 4+3+2=9
9+2=11.
int main()
{
4+3+2=()+2;
return 0;
}
error: expected expression before ')' token
Those idiots forgot to name the function they're calling.
So you give students a problem combining a standard notation they haven't been taught (the algebraic equals sign) and a completely nonstandard notation (using parentheses as meaning "a value to be solved for". IIRC, before algebra when I was in school they used an underscored blank space for that), and the students get it wrong. Big surprise.
This is not necessarily a problem with understanding the abstractions; it's simply ignorance about the notation. Basic arithmetic courses typically have problems like
9 + 7 =
12 - 16 =
where the students are supposed to fill in the value on the right. It's not obvious from this type of problem that the equals sign means "what's on the left is equal to what's on the right" rather than "evaluate the expression on the left and enter the answer on the right".
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.
I'm taking math courses at the local community college. Trying to study at in the libraries there is enough to make one loath humanity. Let's bring our cranky infant to the library so we can watch rap videos on YouTube! Great idea! They also text so much the cell phone might as well be implanted. Get off my lawn!
"I'm not a quack, I'm a mad scientist! There's a difference." - Dr. Cockroach
They were trying to get the kids to understand the meaning of the parentheses, that is to solve whatever is in () before anything else. 4+3+2=(what should go in here)+2 4+3+2=(4+3)+2 They are not trying to get the kids to solve for X or anything.
it is funny that this study comes from a school where there aren't 7 people who can count to 3 so they have 21 guns salutes with 21 people.
mwahahahaha
lose != loose
The problem is that to ensure the school continues to receiving Federal and State funding it needs to meet certain guidelines. These guidelines are measured in the form of standardized tests. The better the test scores are for the school in question, the better funding it receives.
It therefore becomes paramount for the schools that their students get good grades on the test. Therefore it is no longer important that the student understand the answer, only that they get the answer correct. To ensure that the student gets the answer correct, they are forced to memorize the answer to the problems they are given, which are based on problems that will be seen on the exam.
Students that live in our neighborhood are regularly getting Ds on their report cards because they are no longer interested in school. They do the minimum necessary to get by because they are not challenged, in fact they are not even expected to think for themselves. This article is just one example of the result of our current education system.
If you expect your children to be properly educated in our society, it is now your responsibility to do so. Private education, tutoring, home schooling, virtual (computer based) schooling and other atypical forms of education may now be your children's best chance at a decent education. But fundamentally it is now the parents' responsibility, more than ever.
Apparently you missed the part of the article that said "middle school students." We were doing pre-algebra in 6th and 7th grade in my middle school, and we were hardly an elite class or school. If a student had turned in
4 + 3 + 2 = (9) + 2 = 11
for
4 + 3 + 2 = ( ) + 2
they would have asked why you thought the question asked for a second = operator. It clearly doesn't. It asks "what number makes both sides equal." That is how you get late pre-teens to start thinking for algebra.
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
So, when and where did you learn that "( )" meant "an unknown quantity" because I never learned that in any school I attended.
If a "researcher" utterly fails to take into account the developmental progress of the kids they're studying, as well as applying the concept and ages of "middle school" in cultures that don't use the concept, I can't have any faith in their conclusion in particular or their "expertise" as specialists in education in general.
From Jean Piaget http://en.wikipedia.org/wiki/Jean_Piaget
Formal operational stage: from age 12 onwards (development of abstract reasoning). Children develop abstract thought and can easily conserve and think logically in their mind. [Piaget's 4th stage of development]
"Middle school" kids are just entering it. They can grasp an abstract concept. What they can't do very well yet is switch between two of them. Both meanings of = are correct. Finding out that the culture that sells more calculators each year than it has members ends up teaching its kids to use an algebra based on them is kind of interesting, but too critical of itself to get published. Calling it "wrong" is hypocritical, a disservice to the people it's supposed to serve (and that the paper urges to 'read professional journals'), and shows disregard for the history of the "researchers" profession to the point of either incompetence or if purposely ignored to arrive at a conclusion, a violation of ethics.
Middle school is ages 12 and 13 (grades 7 and 8) in my Texas town, Arlington. Two thirds of middle school students either haven't achieved Piaget's 4th stage or finished first year algebra and so haven't been taught that the representative use of = is preferred over the functional use. Two thirds is close enough to 70% to fall within the range of the error (alpha) level considered acceptable by most semi-psychological researchers (p 0.05), who never learn even as professors or editors that an alpha level is what it is and there's no such thing as "acceptable" or not inherent in the concept. Kids grow out of this "inability". Sadly, nobody grows out of statistical misconceptions.
The result here has been seen before in other testing. It is known for being the failure of "new math", which is algebra relabeled in an attempt to make elementary kids learn it. They can't, no matter what it's called.
In other places they have ethnic jokes. In Texas we have (Texas A&M) Aggies. Ethic jokes tend to be the same jokes with the subject changed to suit the audience. Funny sometimes but rarely deserved. Aggie jokes tend to be different -- directly associated to how an agricultural school might view/teach such things as psychology or, well, teaching. Make your own assessment.
"I may be synthetic, but I'm not stupid." -- Bishop 341-B
Call me crazy, but I had a hard time figuring out how they were misreading this to begin with. I couldn't get my head around the idea that they were using '=' to mean "Evaluate", then adding 2. It didn't even occur to me to read it that way until I read through several comments. As a parent, I guess that there are two ways to look at this: 1. The generation currently in school is full of fail. Oh noes! or 2. Well, my kids are going to know math, even if I have to teach it to them myself. If all the other kids are morons, my kids will just have that much more of an advantage. Option 1 is the kneejerk response, option 2 sounds good...but then I stop to think that all those other moron kids get to vote and make decisions in society too. WTF.
Overrated? What, it's not funny enough? Tough crowd!
You can't handle the truth.
I thought he was making a reference to something well-known, but I've been disallusioned.
Intron: the portion of DNA which expresses nothing useful.
That's the problem the students have. My reading has it going like this:
They're taking the blank as a "fill in the answer from the previous part", working the equation from left to right, instead of understanding that the right side is related to the left, and not "part B" of the problem.
This makes perfect sense to me. Helping my little sister with her homework just a few years ago, I would manipulate equations (like moving something to the other side or dividing both sides by two) and she would say you couldn't do that, so I'd have to tell her you could and then give examples that show it was correct. Her teacher didn't get the point that the equation is a whole across, she saw it as two separate things with a symbol in between. But she could usually get the right answers by memorizing the 3 or 4 steps for solving that kind of problem the teacher gave her. But if the problem has a trick in it or isn't formatted right... the students don't know what to do and intuit (incorrectly) how they are supposed to do it.
Comment forecast: Bits of genius surrounded by a sea of mediocrity.
Why are we allowing middle schoolers to use calculators for basic math problems?
Get over yourself.
In the earlier years of school we were forced to do math quickly using math(mad) minutes. Basically a list of problems that you try to answer as quickly as possible. I was terrible at it but other kids zipped through them. I had trouble memorizing the times table and never really did. When I would solve problems I was solving the math problem each time not regurgitating numbers from memory. In order to solve problems it required me to understand the fundamentals of the math. I did terrible in math in the lower grades. Once the math got harder I got better grades and some students that used to do well struggled. I now almost have a math degree. I'm not sure if they teach math the same way these days? If they do I do not believe it's an beneficial way to teach math. I think the minds of young students would be very open to teach the fundamental building blocks of math. When I took discrete mathematics I thought the basic ideas in the class could be taught to children. I found one thing that was hard was changing the way I thought about math. A young mind might be much more open to those ideas.
I just asked a ten year old boy, whose mother brought him into the office. His answer, without hesitation was '7'. And this kid is not in any special program or considered a whiz of any kind. He did not even understand the explanation of the wrong answer. 'Huh? That's stupid', was his response. Out of the mouths of babes.
Did the researchers get their subjects from a school for the mentally challenged and not realize it?
>you could put sqrt(49) in there?
And I would cry tears of joy if I thought for one second that there were teachers out there who would recognize this student as a prize, and not punish them.
But I do not think that. I know what's out there.
-fb Everything not expressly forbidden is now mandatory.
is this the new math and Students don't get the old math?
Jeez. There have always been lots of people who are crappy at math. And there still are. This is like reporting about rain. Yes - there is a general decline in math skills, but there has been a compensating advance in IQ and abstract reasoning. Society requires different skills than it used to - and frankly Math has always been of limited use to all but a chosen few.
There are admittedly problems with our society, but your average Joe messing up symbolic manipulation is not really one of the biggies, and should not be high on the list of things we are worried about. IMHO.
I have a degree in Mathematics Education, and daughter just starting her sophomore year in college. 12+ years of figuring out what they were trying to ask from kindergarten phonics workbooks to high school algebra and pre-calculus. I can't begin to describe how many times when I was asked to help w/homework only to be met with cries of 'we can't do it that way', 'if I do it that way, it won't count', 'that how my teacher said to do it'. wtf are they trying to do and who came up w/these cockamamie teaching methods?
Is it really that confusing? The problem is the lack of logic and understanding of equality. It shouldn't matter what the equation looks like, if one side equals the other, one side equals the other.
This is where I blame computers for making math a little more difficult.
5 x 2 = x x 1 is a real PITA to go through when you type it out, even if you wrote it as 5 x 2 = X x 1, it looks like crap.
5 * 2 = x * 1 works better, but no one outside of computers uses * for multiplication, which is a shame, because it doesn't look like a #@$@ing letter!
Using the highest level of the order of operation is NOT a good replacement, even if it can be deciphered fairly easily.
(5 + 2) x 2 + (4 + 1) = (() + 1) looks like crap as well.
So, either solve for y, or use * to multiply.
For teaching to the test. I'm thankful that I came up through schools before all this crap was put in place. Learned the algebraic method etc. My involvement with computers reinforced the meaning of the = sign and then the four functions (+, -, /, *) as well as modulus and a few other statistical goodies in between.
I suppose a calculus integral symbol or even the derivative dx/dy would throw em'.
... that you find the problems that stupid people are facing to be much like your own.
But look at it this way. Mod points come and go to all of us, but stupid is personal trait.
Mit der Dummheit kämpfen Götter selbst vergebens
On a worksheet ( ) would be a normal way of saying fill in the blank a precursor to learning about variables. Kids are taught fill in the blank problems first and the ( ) is used as a blank space since in when written by a 5th grader (and many adults) _ tends to move toward the center line thus getting confused with -. It may even be confused with ~ if the writer had to much to drink last night.
And even before that, BASIC used LET X = 1, which wasn’t terribly confusing if you ask me.
Alexander Peter Kristopeit bought his basement from his mommy for one dollar.
Hrm. I see it like this:
4+3+2=x+2
4+3+2 = 9
so, in order to make the left hand EQUAL to the right hand, I'd need to make x = 7
so: 4+3+2=7+2
Both sides of the "=" are 9, therefore equal.
There's nothing wrong with 5+2=7=9-2 as a mathematical statement. It's not a traditional equation, but it's fine as a statement.
Please consider this account deleted, I just can't be bothered with the spam anymore.
I think the problem goes deeper than notation. From what I can see, students are too reliant on blind procedure to get the "answer". They often lack logical skill at the most basic level, missing concepts such as argument, truth, and falsity.
Consider a contradictory system of linear equations, say,
x + y = 5
x + y = 2
Of course, many students will fall back to the graphical approach with this, which is fine, except that I'm not sure they understand what graphs actually mean (e.g. the set of points which make an equation true). They will say there is no intersection, and they have equated the intersection of lines with "the answer", so there is no "answer". But do they understand that each line is the set of all points (x,y) that make each equation true, and that the lack of an intersection means that there is no single point that makes both equations true? Often they don't.
And if they follow the subtraction procedure, where they subtract the second equation from the first equation, then they will get the contradictory equation
0 = 3
Many of them will be flummoxed by the meaning of this. They may make up non-sensical answers, or they may actually get the answer "no solution". But even when they get the answer right, I am not convinced that many actually understand the logical justification for what they say. I suspect many of them do not understand that the above statement is a contradiction.
I think that one way to improve this is to instil in students the habit of logic at an earlier age. I think that teaching some basic logic at the ages of 12 or 13 would improve this. We need to teach these children that there is such as thing as truth and falsity, and that they need to back up what they say with some reasoning. We need to get them into the habit of giving their logical reasoning. I think this could help break the cycle where students learn that the only point of doing a question is to get "an answer" in order to get marks, as opposed to actually understanding a concept.
This and no other is the root from which a tyrant springs; when first he appears as a protector - Plato (423 to 327 BC)
For example: say that I had a dollar bill and 4 quarters, would those be equal? The answer is both yes and no... The 4 quarters and dollar bill have the same economic purchasing power, but the properties of each are different, including chemical makeup, density, look, etc. So they are in fact different, but the same. You could also argue that a glass of water is less than a pound of gold, but without the context...like being in the middle of a desert...you cannot rationalize the equality or inequality of the them. School doesn't tell the stories of the problems very well; schools now lack the excitement and curiosity provoking that needs to occur to bring kids attention back to learning and separate themselves from their video games, cell phones, and Facebook, etc.
Is that the students should be taught to THINK instead of just memorize.
Had the problem been 4+3+2=z+2 (Solve for z), would you also argue that it "Should be x, cause it is always x"? How about 4+3+2=u+2?
How about $4 + 300 cents + 2x($1) = ($$$) + $2?
Real life problems don't come in a "X marks the spot" form.
As for who is to blame... well, that has been determined eons ago:
Ssu-ma Ch`ien gives the following biography of Sun Tzu:
Sun Tzu Wu was a native of the Ch`i State.
His ART OF WAR brought him to the notice of Ho Lu, [2] King of Wu. Ho Lu said to him:
"I have carefully perused your 13 chapters. May I submit your theory of managing soldiers to a slight test?"
Sun Tzu replied: "You may."
Ho Lu asked: "May the test be applied to women?"
The answer was again in the affirmative, so arrangements were made to bring 180 ladies out of the Palace.
Sun Tzu divided them into two companies, and placed one of the King's favorite concubines at the head of each.
He then bade them all take spears in their hands, and addressed them thus:
"I presume you know the difference between front and back, right hand and left hand?"
The girls replied: Yes.
Sun Tzu went on: "When I say "Eyes front," you must look straight ahead.
When I say "Left turn," you must face towards your left hand.
When I say "Right turn," you must face towards your right hand.
When I say "About turn," you must face right round towards your back."
Again the girls assented.
The words of command having been thus explained, he set up the halberds and battle-axes in order to begin the drill.
Then, to the sound of drums, he gave the order "Right turn."
But the girls only burst out laughing.
Sun Tzu said: "If words of command are not clear and distinct, if orders are not thoroughly understood, then the general is to blame."
So he started drilling them again, and this time gave the order "Left turn," whereupon the girls once more burst into fits of laughter.
Sun Tzu: "If words of command are not clear and distinct, if orders are not thoroughly understood, the general is to blame.
But if his orders ARE clear, and the soldiers nevertheless disobey, then it is the fault of their officers."
So saying, he ordered the leaders of the two companies to be beheaded.
Now the king of Wu was watching the scene from the top of a raised pavilion; and when he saw that his favorite concubines were about to be executed, he was greatly alarmed and hurriedly sent down the following message:
"We are now quite satisfied as to our general's ability to handle troops.
If We are bereft of these two concubines, our meat and drink will lose their savor.
It is our wish that they shall not be beheaded."
Sun Tzu replied: "Having once received His Majesty's commission to be the general of his forces, there are certain commands of His Majesty which, acting in that capacity, I am unable to accept."
Accordingly, he had the two leaders beheaded, and straightway installed the pair next in order as leaders in their place.
When this had been done, the drum was sounded for the drill once more; and the girls went through all the evolutions, turning to the right or to the left, marching ahead or wheeling back, kneeling or standing, with perfect accuracy and precision, not venturing to utter a sound.
Then Sun Tzu sent a messenger to the King saying: "Your soldiers, Sire, are now properly drilled and disciplined, and ready for your majesty's inspection.
They can be put to any use that their sovereign may desire; bid them go through fire and water, and they will not disobey."
But the King replied: "Let our general cease drilling and return to camp.
As for us, We have no wish to come down and inspect the troops."
Thereupon Sun Tzu said: "The King is only fond of words, and cannot translate them into deeds."
After that, Ho Lu saw that Sun Tzu was one who knew how to handle an army, and fin
Mit der Dummheit kämpfen Götter selbst vergebens
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.
The problem here is the way the question is asked. It should be 4+3+2=X+2, solve for X. The students are not so likely to be misunderstanding the equals sign as the odd paradigm of using parentheses to denote a variable. They're forced to try to guess what the question is looking for, and it's not so odd that some of them got it wrong. The chain of expressions connected by equals signs is exactly what is generally used when reducing an equation, so that's a perfectly valid use of the equals sign, and a reasonable interpretation of what is being asked for, although it should be 4+3+2=7+2=9. Still, it seems to me that maybe Capraro is the one who doesn't understand the equals sign.
Math education in America is pathetic. I went through my nephews High School textbook and there wasn't any MATH in it. There were lots of pictures of butterflies and "Why are we learning this?" columns and the whole thing looked like it was designed to be entertaining, rather than educational. The math was an afterthought, with hardly any problems, no explanations of those problems or how to solve them, and no answers. I was stunned, especially when I learned it was written by four math professors.
There is some argument, of course, that this is on purpose, and that we fail our duties to educate our children because an educated populace would be a danger to those in power. I'm not prepared to accept that, but I do think we've completely failed in our duty, and the uneducated masses of today is evidence enough of that.
My father has a saying, "There's no teaching if there's no learning. Until there is learning, you aren't a teacher, you are simply a presenter". I think we have far too many presenters, and not anywhere near enough teachers.
What exactly sounds weird about Ohioan and Wyomingite?
More to the point.. why would those sound any 'weirder' than people from TexaS being TexaN or people from Puerto RicO being Puerto RicAn, while people from Massachusetts aren't Massachusettan but Massachusettsan?
Not to mention Connecticuter.. cuter? Surely for pronunciation that should have been Connecticutter?
At least Ohioan makes it more clear it's somebody from Ohio than New Mexican does for somebody from New Mexico.. that should have been New Mexicoan as well.
Then again, I'm from The Netherlands, or Holland if you prefer, but you English-speaking folk insist on calling us Dutch.. so maybe I'm just used to these sorts of shenanigans ;)
I believe I understand what an "equal sign" means, but I've never really questioned it.
In math I've come to understand everything as a... series of truths, such as
1+1 is true to 2, therefor 1+1=2
Then when I took pre-algebra for the first time in college (I had some serious problems in grade school and let's just say my educators failed to provide for me) the importance of the "equal sign" became more significant. Instead of trying to find one truth I might have had to find many, but I was never really removing or adding anything that wasn't there before.
There was when I realized that my duty in establishing a connection on two "sides" of an equation had nothing to do with what I really did, it was either already true or not true or one side was not yet determined.
So really the math itself changes nothing in the way I understood cause and effect and what I could do about it, you're just seeking a different expression of what's already there.
I don't know if that makes any sense but it seemed like an awfully important thing to me at the time.
"Most people, I think, don't even know what a rootkit is, so why should they care about it?"
Even Graham, Knuth, and Patashnik use "=" instead of the is-an-element-of symbol (Hey Slashdot, would it kill you to support Unicode x2208?) in their use of Big O notation. In English, this would be a confusion between identity and attribution.
Reminds me of some funny answers given by some students:
Q: Expand (a+b)^2
A: ( a + b ) ^ 2
and
Q: x^2 - 2 = 14 find x
A: ^here it is.
So cars with automatic transmission are only imaginary cars? Strange, they seem to fulfil the role of a real car quite well.
Now shut up with your stupid synchronous transmission! :-)
The Tao of math: The numbers you can count are not the real numbers.
X is used for this, not (), not ?. X
Before you question someone elses math, learn it yourself (not talking about parent, but the article submitter.
There is a reason we use X and the reason is because it doesn't clash with anything else.
() is for grouping.
Really, calling kids out because they don't understand = and then not getting X yourself... how is that for irony.
MMO Quests are like orgasms:
You may solo them, I prefer them in a group.
Whooops, that would be me there
Would you like a slice of toast?
A 7 within 5? I've never seen such a number. What does it mean to write a 7 inside a 5?
The Tao of math: The numbers you can count are not the real numbers.
possibly even the people doing the study...
I say this as a public high school math/science teacher in the US.
In older and more traditional texts - the x or "empty square box" or the fill-in-the-blank-underline is readily found as the missing variable. It is only textbooks/materials from the last 5-10 years where that "( )" *@&#^&@%#^!!!#! has started to appear in the lower level, and now, the more advanced high school text books.
When i first encountered it - I was totally confused by the question (like many here on slashdot). However, having gone to mathematics conferences, talking to people in the textbook publishing field, and the actual authors of some of these math texts - I can only conclude the following:
1) A lot of the editors of these books are flatout clueless. Authors come up with the text materials, and insert blanks, question marks or underline in place of variables for the lower level/basic texts. However, from text translation from one form to another (example material formatted on a Mac and then opened on a PC) you get the occasional random character because of whatever reason (Yeah, I'm that old where I can recall this being a typical problem!) between formats. The editors see the strange characters, or heck, even a question mark, assume it's a boo-boo and have changed it into the whole parenthesis insanity. Since these same editor types usually don't just edit for one text - practices like this get transferred over from one text to another.
2) from my experience talking to elementary / middle school teachers who teach math... the vast majority of them hated math as a topic with a passion. And when asked why - it comes down to the fundamentals of not really understanding the material. I have pointed this out again and again - when you have people teaching something that they've hated and/or don't truly understand - you are not going to get students who will grasp the material and understand it! These math teachers do not understand that the formatting/question method is not in the best interest of the students - and don't question it! They do enough to get by to get the kids to pass whats on required on the state exams - and they're done. So between a bad model and bad modeling - yes, you will get students who don't get it!
it's just aggravating to me that when I teach chem/physics - a lot of times, I also have to teach what i call "Algebra-Zero" to show the kids what some of the things that they do is totally wrong, and how to it's really not that bad...
Nope. That's not what it means at all. The apostrophe (that's its name) is used to denote the possessive e.g. JeremyP's post. It's also used to show that you've missed out some letters, as in "you've" or "it's" or "couldn't". So technically "math's" is correct, but nobody ever writes it like that.
All I want is a secure system where it's easy to do anything I want. Is that too much to ask ~~ Randall Munroe
You have a very valid point!
Parenthesis should never be used as a substitute for a variable. Parenthesis have a well defined meaning in mathematics. Using them as a substitute for a variable introduces confusion when a child is introduced to their proper use. There are many other symbols that can safely be used but avoid those that have a common mathematical meaning. Of course, in higher math, almost every symbol has a meaning, just avoid symbols that are commonly used.
=== to test equality
I think ADA was := [ http://groups.engin.umd.umich.edu/CIS/course.des/cis400/ada/array_summation.html ]
Java, C, and C++ use ==
Shell script, eq. ...If I am remembering correctly :)
Uh, Linux geek since 1999.
So they don't understand NULL. That's acceptable. 70% of people saw the equation had a memory leak and carried data over into the null that just happened to be the last equation they saw. That happens all the time to a computer program that's not properly debugged. I'm more worried about the 30% of people who saw null and created X. Who are they to just randomly initialize variables to catch an exception that they didn't know was going to come there way any time soon.
This is the real order of how it was learned, not which order they were teached. :: R->R
1) numbers
2) addition/substraction
3) expressions (1+2+3+4)
4) variables x,y
6) equals sign x=10
7) functions f(x) = x+10
8) equations x+10 = 20
9) booleans / and/or/not
10) cartesian products (thanks to programming with structs in C language)
11) types f(x)
12) solving equations x+10=20 => x+10-10=20-10 => x+(10-10)=20-10 => x = 20-10 => x=10
13) implication
14) logic (forall/exists)
15) substitution
16) equivalence class/equivalence relation (thanks to beta reduction/lambda calculus)
17) subsets
18) identity vs equality
19) set theory (intersection)
20) diagram commutes
21) equalizers
22) pullbacks
23) limits
24) inverse image
I've omitted anything that I don't think are related to the equals sign problem in the topic. I'm probably still missing some important parts of the problem. It's really quite complicated problem when you start to think about it. You can use a lifetime studying the problem and it still can give you lot's of fun when you learn new things about it. And this is just one path how you can learn about it, there are probably many other nice ways to learn about it.
Major university math/engineering department chairs, well known scientists, and pretty much anyone with science or math degrees are protesting the most harmful thing to happen to math(and education) since No Child Left Behind. That horrible thing they are protesting is Everyday Math, which is continuing the destruction of math education in the US one bright young mind at a time. This article touches on memorization, but EM goes beyond that, since you don't even learn math mastery, and this is an elementary school program.
...any argument about the misunderstanding of the equal sign from someone who misuses the word "is".
From TFV: "One of the bigger issues is, is that..."
No, the food on your MS Office ribbon is because you're an incredible messy eater. Stop yelling at MS Office during lunch, it's probably scaring your coworkers.
Speaking as someone never that great at maths...
4+3+2=( )+2
WTF is that meant to mean? My issue is not the = but rather the ( ).
Surely the question should be:
"If 4+3+2=x+2
then what is the value of x?"
Or, if you've not gotten into the whole letters representing numbers thing yet, follow the textbook staple of the underline for filling in the blanks, i.e. "4+3+2=__+2".
If I see ( ) my assumption is there is meant to be a self-contained operation going on there, it's an instruction of the order of calculations. Everything that happens inside the brackets either has to be calculated first before you can calculate the rest, or you need to first rearrange the formula.
x(y+z) means you either add y+z before you can multiply result by x, or rearrange formula to xy+xz.
The students probably misunderstood the parenthesis as if it was "(4+3+2=___)+2=?". This probably still isn't right but there's a logic to it.
Oh and I failed by Higher maths mock exam. My dad's friend, an engineer, started tutoring me and I got a good B (67-69%, probably scraping me into the upper quartile) in the end. The tutoring did it's job within the first hour. We went through my answers where he was immediately surprised I'd beaten some difficult questions. Turned out there was one thing I didn't know about which repeatedly gave me no chance in certain topics, as I recall we wondered if I'd simply been off sick the day it had been taught. 10 minutes of discussion and a few examples later it was learned and the next couple of lessons were clearly unnecessary and we decided to stop.
If the mock papers hadn't been marked by a pool of teachers, and/or our class size had been less than the 30-something we had, then the teacher probably would have picked it up himself.
Because math teachers in America suck.
FTA:
The problem is students memorize procedures without fully understanding the mathematics, he notes.
Thank you American math teachers for teaching me how to memorize procedures for 12 years instead of, you know, actually helping me understand what it all means. Most importantly, thanks for not showing me how it applies to real life.
I have an RPN calculator. There's no equals sign on it!
Have gnu, will travel.
Did they test to see if 4 + 3 + 2 = 2 + __ worked better than 4 + 3 + 2 = __ + 2? I would think that the root problem is that the order they used looks too much like (4 + 3 + 2 = __) + 2.
When I used to help him with his homework I was often appalled at how poorly written his mathematics books were. Often there were problems that were impossible to solve because there were mistakes in the figures or wording. I would usually write a note in the margin explaining why the problem was insoluble. No feedback from the teacher, ever. I personally found the Singapore Math to be counter-intuitive. That's a pretty bad foundation for later learning
Seems to me the majority of teachers are actually stupid and merely get by on having the teachers' answer book. They barely comprehend the basic mathematics themselves. They have too much work to do to do their jobs properly(!)
The other thing that has always annoyed me deeply is this phonetic spelling thing. Parents are told not to point out spelling errors (because it might hurt little Johnny's self esteem!) and that phonetic spelling is acceptable in the early years of school. My question then is - when is incorrect spelling no longer acceptable? At retirement age? I think this nonsense goes a long way to explain the very American their/there/they're debacle.
http://www.acetonestudio.com
An anonymous scalar or something like that? Seems vaguely perlish.
I'm from Germany; we had calculators in school. However, we only had them for the last few terms. Essentially, we were allowed to use calculators that could solve problems not in the current term's curriculum. Linear algebra is not about adding and dividing numbers or calculating random square roots in your head, thus we were allowed to use tools to do that for us so we could focus on the problem at hand.
Of course we could have stuck only to problems that break down into integers and simple fractions but the real world doesn't always work like that. It's much better to have a graphing calculator that enables you to tackle some more realistic problems. (And no, they weren't useful for cheating. We had to use school-supplied calculators for our tests.)
USE HOT GRITS WITH STATUE OF NATALIE PORTMAN (NAKED AND PETRIFIED)
I can understand that trying to refer to an object that doesn't exist distinctly, as if it did, causes some confusion.
That's just an imaginary / obscure / weird / cute way of writing 1
You could write it as (1/9) * 9 it's exactly the same amount of obfuscation. Since a digit D divided by 9 is what represents a digit repeating in Base 10.
There's really no such number as 0.9999999...., as in there is no rational that expands to "9" repeating, when expanded to a decimal fraction using basic arithmetic.
So it shouldn't come up too frequently in basic arithmetic.
It's pretty obvious that the fraction 9/9 does not yield a repeating decimal, basic arithmetic operations expand that to 1.0, not 0.99999....
The only way you really arrive at a repeating 9, with only basic arithmetic, is if you tried to manipulate another repeating decimal, eg If you took 0.1111111111.... and tried to apply arithmetic multiplication by 9.
In basic arithmetic, this is invalid, you cannot really expect to get the right answer if you multiply or divide repeating decimals. You get an object that doesn't exist distinctly in basic arithmetic, because multiplication of a repeating decimal gives you an undefined result.
Much in the way you get an ambiguous result if you try to supply "0" as the right operand to the divide-by operation.
I agree completely. See my post re Singapore Math. My mathematical abilities are nothing special at all, but the way these books for elementary school are put together is awful. There is a reason for standard notation and syntax and throwing that out to make it easier to understand with no standard way of doing it makes it even more confusing. I expect that my sons will have an even more difficult struggle with mathematics than I had.
http://www.acetonestudio.com
I had no idea the intent of the parenthesis was to indicate a variable.
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The question lacks context. If the idea was to use ( ) as a variable (say, "x"), then the question would have to say, "Provide the appropriate value to go in the parenthesis to solve the problem".
A work that expires before its copyright never enters the public domain and thus enjoys eternal copyright protection.
looks like my post on Singapore Math got thrown in the memory hole. What's up with that /.?
http://www.acetonestudio.com
>Am I the only one who absolutely DID NOT understand your answer? How do you go from: 4+3+2 = 9 to: ( ) + 2 = 2 ? It makes no sense.
I summed the left and right hands of the equation separately.
The left side was 4+3+2, which equals 9
The right side was ( ) + 2, which I interpreted as 0 + 2, which is 2.
A work that expires before its copyright never enters the public domain and thus enjoys eternal copyright protection.
I have no authority to cite. I simply did not understand that ( ) was to be interpreted as "x" (or some other variable).
Thus when I see ( ) + 2, I interpret this as "nothing plus two". Parenthesis are operators, and the operators are operating on nothing, thus nothing is being added to two.
A work that expires before its copyright never enters the public domain and thus enjoys eternal copyright protection.
If you use a system that does mathematics, like MathCAD, you run into the ambiguities of the equal sign. There are multiple meanings, and in most automated systems, there are separate symbols for them.
Then there's operator precedence and scope. In advanced mathematics, this can drive you nuts, as papers often introduce new notation without being clear on operator precedence. There's also the macho thing in mathematics of using as few parentheses as possible. Variable scope issues in math are awful. I once encountered a symbol in a book on nonlinear differential equations which was defined in the previous volume of a two-volume set.
This isn't a trivial issue for new students.
Math was introduced to me like this: arithmetic (and a little geometry) -> pre-algebra -> algebra -> basic geometry -> basic algebra -> more algebra -> trigonometry -> calculus. I feel like one of the fundamental problems with my education was that it seemed like arithmetic was fundamental, but geometry is just as basic. I think the real heart of the issue to me is logic. I believe that math and logic are important enough to our education (and increasingly so) that the two should not be conflated, but taught separately. I think part of the reason that they are mixed together is the apparent difficulty of logic. Most adults don't seem to be able to discuss it, even though they use it constantly. Teaching it to children at a young age seems crucial to me. I think that kids ought to get a separate logic book. I haven't worked out the particulars, but I'd say that issues like how == works would be covered, maybe as early as ... third grade? By 6th or 7th grade, I think most kids could have a handle on types of proofs, the if-then stuff, the rudiments of sets, and AND, OR etc. Obviously, the way that such a class could be 'sold' to parents, educators, politicians etc. would be as a computer class, but I think it would be equally (if not more) important as a foundation for higher math.
I would have much preferred a logic class to the daily worship sessions I got at my tiny private elementaryand middle school.
Too idealistic? Too high of expectations? I feel like for America right now, it's sink or swim.
I guess I can't argue with you about the overall success of the new math, but I have very fond memories of it. I was in third grade in 1965 and well remember the exercises in counting in base 2, base 8, and base 12 using popsicle sticks. I well remember the lightening flash of understanding: "Oh, there are numbers, and then there are the representations of numbers, and each number can have many representations". Then I switched schools, and math became an exercise in tedium until I got to college.
The "new math" taught in the 60s was designed by mathematicians. The new "new math" (AKA Discovery math) being taught today was designed by educational theorists. Maybe the new math of the sixties was only suitable for students who were going to go on to become mathematicians, computer scientists, or physicists, and only when taught by teachers who really understood the point of the material. From what I've heard of discovery math, it's only suitable for people who are never going to do any math, and who will rely on calculators for simple arithmetic.
Looking at the problem isn't the correct answer 4+3+2=(4+3)+2 or 4+3+2=(3+4)+2 The problem with parenthesis are they have a mathematical meaning already. They determine the order of precedence.
+1 bad pun.
(oh sorry, "bad pun" is redundant.)
Seems to me like these kids should spend some time at Khan Academy.
It is no measure of health to be well adjusted to a profoundly sick society. - Krishnamurti
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
Fortran (at least up to Fortran77) uses "=" just for assignment. For comparisons we have to use (x.eq.y). The system works pretty well, and much simpler than C. However, typing .eq., .ne., .gt., .and., etc at every logical group is a royal pain in the ass.
...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!
4+3+2 is not equal to 9+2.
Which is why 70% of the kids added "=11", which is correct.
They simply underestimated the importance of the = sign, and were simply following calculator math: "4+3(7)+2(9) = 9+2(11) = 11"
I'm sure every one of them thought "That's an odd way of writing the question, and there's no place for the final answer! Oh well".
If they don't know how to solve for X, of course they're going to have a hard time with anything that tries to get them to solve for a retarded substitute for X. Just have them solve for X, damnit!
Security is mostly a superstition... Avoiding danger is no safer in the long run than outright exposure. - Helen Keller
BTW, the example equation (function?) of 4+3+2=( )+2 is weird. It would be more proper to write it as 4+3+2=X+2. Anyway, I think I can see why someone would make the example mistake. It's a confusion of the order of operations. If someone thinks + and = are on the same level (like + and - is) and does it left to right, then an ignorant person would see the equation as [4+3+2=( )]+2. Therefor, they replace ( ) [or x] with 9, and then tack on the +2 and solve/compute the new 9+2 equation.
By the way, on a related note, is there any proper way to add on operations to an equation that you're solving or simplifying live? For example, when trying to figure out the proper placements for the items on an image, you may be working with a bunch of dimensions and considerations. As you muddle through it on paper, you write 73-27=46. Then, you realize you need to subtract some more, so you continue writing and get 73-27=46|-24.75=21.25. Then you realize you need half that, and so you write some more and finally get 73-27=46|-24.75=21.25|/2=10.625. Of course, if you were to present this to someone, you would write it as (73-27-24.75)/2=10.625, but a train of thought (or at least my train of thought) doesn't always think that far ahead during computations. So, I use the bar | notation above, or sometimes use new lines, but is there any "proper" way to do it?
Exactly, but I don't think this problem is particularly new, or maybe I was just "ahead of my time" in the ignorance department. After getting excellent grades in all subjects through elementary school, I hit my first skid in 7th grade pre-algebra, and it was because of this problem. I spent much of a semester failing or barely passing tests simply because I didn't get that an "equation is a whole across," as you put it. You can only get so far when you have a fundamental misunderstanding of what "=" really means. Once it was finally explained to me, I did fine and my grades recovered. But it shouldn't have taken my math teacher (who was a putz) so long to figure out what was going wrong. Sure, sometimes students are lazy, but just as often teachers are inattentive and unmotivated. If you can't get the basic principles across, you aren't doing your job.
"No live organism can continue for long to exist sanely under conditions of absolute reality;..."
I hope you showed her that "moving something to the other side" is actually shorthand for adding its opposite to both sides and simplifying.
I don't get why so many people here are confused by the brackets. To me, an equal sign simply means the left part and the right part of the sign are the same thing. Anything that is not a number or an operation is simply variables.
My first thought when I see that 4+3+2=()+2 is to remove the 2s and arrived at 4+3=(). I had more difficulty understanding the 4+3+2=9+2=11 part, since it makes no sense whatsoever in the presence of the equal signs, until I realized that it makes sense for people that use calculators.
I agree that the problem stems from the use of calculators in school, where people would equate the equal sign as an operation (on par with +,-,etc).
This is a good reason to bring back the old RPN HP calculators and mandate its use in school. Those things don't have an equal sign and basically forces you to perform math properly.
Wow...I agree. I think the problem here is not that the students don't understand the meaning of the "=", but that it is stupid to use a empty pair of brackets (with no well defined mathematical meaning) as a substitution for a variable... Is this really how single-variable-algebra is taught in US high-schools?
The way to academic success: always tell people that you think their problems are crab and that you couldn't care less..
SMSG, or Some Math Some Garbage, as we clever 7th graders used to call it back in the day.
Are you suggesting that Microsoft employ fast food operatives as interface designers? Hmm, you might be on to something.
//to do: invent jibe about open source interface designers
Confucius say, "Find worm in apple - bad. Find half a worm - worse."
Abolish? Infix is appalling, and I use calculators very sparingly, like 5 or 10 numbers added together per week.
It has no purpose, unless you want to perform exactly one operation on exactly two numbers and then never do anything again. If that's what you're doing, infix is A-OK.
Comment removed based on user account deletion
I am not an engineer and understood the problem immediately. Calculating and checking the solution wasn't required, for it was an additional trivial step, that I would only undertake if I needed to know the value of x. Understanding the mathematically precise version (according to the engineers) took longer, because it had additional information that was not required. I had to review the problem to check my understanding of it was correct, for aside from being longer, it used English words surrounded by brackets as additional instructions in a problem that originally only used mathematical symbols, if not precisely correct ones, according to the engineers. (Solve for x) should be obvious to anyone not programmed to only think one way, although pointing out that this was incorrect is perhaps correct. While it is important to teach things using absolute precision from time to time, this should be done in moderation. I suspect that by forcing and enforcing the use of absolute precision, the ability to subsequently think in ways not absolutely precise may be reduced. This might be useful if you want to modify the way people think in different nations. Humans are not computer programming language compilers. Always expecting and enforcing the perfect use of a single particular method when expressing a problem that is intended to be solved prevents alternative methods from being used or tried. Optimization requires us to constantly create and try alternatives, not to mention attempting to both shorten and simplify things. Choosing methods of expressing problems based primarily on teach-ability may prevent students from learning for themselves. Choosing methods of expressing problems based primarily on teach-ability probably prevent the teachers from learning as well, which in turn may affect the students ability to learn from them. These observations may not apply to this particular example. I am not a teacher, so these observations might be more appropriately called guesses or assumptions, or bullshit. If I apply a mathematical method of explaining them I think they might be called explanations of possible interpretations. I don't know which specific "completely precise" definition of the word equation you are applying to what the author of the summary termed a "problem", so I certainly don't know if your claim that the equation lacked the precision of mathematics is correct. I do know the author of the summary also used the word 'work' and the word 'procedure'. I have now read many differing definitions of the word equation and am little wiser. Unfortunately, each explanation I can understand and interpret in many different ways. If I could improve mathematics myself, considering myself a well-intentioned perfectionist, I certainly wouldn't waste time being draconian with minor variations in the presentation of simple primary school algebra problems. I would create new words for numbers, ensuring they were all single-syllable up to 1,000,000. Or as far as we can, at least. To 100 would be a start! To have to resort to saying twen-ty-two in three syllables is stone age, at best. And how many times can 2 go into 11 ? (the time spent saying) Does this equal three ? (or not) Further, I would also create new single syllable two, three or four letter words for every other thing associated with counting, calculation, and mathematics. This is a worthy job, and no doubt it would give engineers much joy to be pedantic about the precise pronunciation of the expression of the new mathematical terms. Reducing the time spent saying and expressing numbers and formulas would probably be a far better way to improve mathematics and the teaching thereof. This would also achieve that which those engineers and teachers who program would perhaps most appreciate - a distinct separation between language and mathematics. Subsequent to this we could legitimately expect others to adhere to absolute perfection with no margin for error within what would then be clearly only ever a static and unchanging mathematical domain. I wouldn't join you there. Getting back to my original point, maybe Americans would be better at understanding the concept of e-quals if it was simplified to just "eh". I know it would have helped me. Here's hoping I wasn't vague.
The reason the expression 4+3+2=()+2 is confusing to you ol' slashdotters is because you don't really understand what the equal sign means. You also apparently don't really grasp the concept of assignation. The equals sign simply means "is equivalent to" or "can also be written as". 4+3+2 can also be written as ()+2 which can also be written as 9. Assignation is denoted by a similar symbol made up of three parallel horizontal lines. Don't confuse computer science with mathematics.
This is not the rule, nor are several other popular explanations I've heard. Unfortunately, the end result is that I hear many people in America saying things like "Aerosmith are playing ..." specifically when talking about bands. My guess is that this is because the hipster types who do the most writing/talking about bands have a hypercorrective urge to appear cultured by conforming to fictional linguistic oddities.
For the real rule (well, description of common use before the fauxrule took hold), see http://en.wikipedia.org/wiki/American_and_British_English_differences#Formal_and_notional_agreement .
There is an enormous amount of complaining about notation here...funny. Get out of your engineering/prorgramming mindsets, it makes you look just as bad as the kid who does not know what '=' means in a mathematical statement. Look, we have the statement 4+3+2 = ( ) + 2. First of all, focus in on the fact that the person reading this statement is not someone who knows things such as programming, or would even try to analyze the idea of 'nothing in the parenthesis - must be a zero'. We are looking at kid, who has numerous times already seen the problems of the form 5 + 6 = ( ). The language is not new to a 1st graded, 2nd grader, 3rd grader...and you know what, it is not new to any of you either, you are just hung up on the way you learned mathematics later in high school and then in college. You are even hung up on the symbol 'x' which carries alot more extra meaning than the empty space. (Think about the problem: fill in the blank, "I _______(eat) too many cookies for breakfast and now my stomach hurts" - I am sure most of you have no issues realizing that the problem is asking you to write down 'ate', not 'blame my sister for having eaten'). Return to the original issue at hand. The schools in United States are focused on having kids learn to solve specific format of the problem. Mathematics are not taught as a set of rules that can be applied universally. Rather - for every problem there is a specific order of steps you follow. Change the problem slightly, and everything goes downhill, students try to do stupid things. Heck, in high school I had a teacher who would give word problems with more information given then necessary to answer the question. We had a straight-A student stumped over it because he couldn't figure out what equations he had to invoke to use all the data! Education here does not push understanding the problem - it pushes memorizing solutions. So the kid in the OP's example sees '4 + 3 + 2 = ( ) ...', and ignores the rest of the statement. It's more of 'I know what to do with this set of symbols!', not 'what does this line say?' Someone here calls it a poorly debugged complier - but a compiler is worthless as a human being. Educating to understand the meaning of each individual symbol should produce students that have no issues understanding such new statements 99% of the time.
What I am trying to say: we should be trying to teach a language. There is no difference between "4 + 3 + 2 = ( ) + 2" and "sum of four, three, and two is equal to sum of something and two", and the remainder of the question is just "fill in the blank", which someone taking a test is keenly aware of. Understand the meaning of symbols - parsing is trivial for the human brain. Don't understand the meaning of symbols - and you are an outdated compiler.
On top of this: you folks are so obsessed with the variable... many times to date when teaching kids I found that having an empty space for a number is much easier than an x. Kid with no prior education in equations will attempt and likely succeed to solve an equation with variable listed as an empty box (mind you, I focus on teaching my students to understand individual symbols as words in a sentence). Put in an x without prior explanation - things go downhill fast. The variable itself is an abstraction that takes quite a bit of effort to understand, and many many people never understand at all (going through years of even college level education after).
Sets use curlies.
Someone had to do it.
In my experience (over 15 years of tuition and teaching in mathematics and physics to all ages and abilities) people's 'understanding' of such mathematical statements varies wildly. I have had many students who could answer the question correctly and yet fail to explain (correctly) how or why they arrived at the answer. For me that they have not learnt any mathematics there but simply how to get by. I have also had students who could answer the question correctly but their 'explanation' was technically incorrect. In fact this is extremely common with such algebraic statements and most of it is down to how they were taught maths at a very early age. In my opinion most of this groundwork is laid at Primary school age (up to say age 11) and sticks with them for a long long time unless those misconceptions are tackled by an expert. The vast majority of primary school teachers (at least here in the UK) have no formal mathematical qualification (by that I mean at least a decent grade at A Level) and a tiny tiny percentage have any form of mathematical degree (that is before we consider the quality of that degree itself and their teaching) and yet they are responsible for getting this concepts down and clear and laying the foundation for potentially another 10 to 15 years of advanced maths. I have seen many of my A Level students (ages 16-18), some of whom were getting good grades, fail to truly understand or explain expressions similar to those above. It is quite important to tackle these ideas at an early age, and it is surprisingly easy to do as well. Sometimes it can be as simple as not ever explaining what the "=" actually means (or later the difference between an equation and an identity). Sometimes it is an over reliance on just one or two examples (when I introduce algebra I am sure to use a variety of letters, symbols, pictures and examples rather the age old standard "x" or "empy box"). Sometimes it is born out of poor technique ("move things to the other side" - no such mathematical operation). Sometimes it is as a result of the teacher failing to explain what they are doing and why (which can be difficult for abstract mathematics but the maths we want a 6 to 16 year old to learn can be firmly rooted in real life analogies and examples as long as it is made clear they truly are just a crutch to help the student understand).
Well I had no idea learning what the = sign means (20+ years ago).
Either the school's now cant teach anymore or the current generation dont need to understand what the = sign means.
How long until books are written in sms text speak (l33t speak even)?
For years I struggled with the greater-than/less-than signs because I didn't realize that the limitation of reading it from left-to-right was in effect. I often wondered what would happen if you went from right-to-left, would you have to change it from greater-than to less-than (or vise-versa)? I don't know how, but I was never ever able to convey that question to any teacher well enough for an answer. Nor did they ever explain it so I understood. I was so furious at myself years later when I realized (during a programming class) the simplicity of it.
I'd always thought that equality was simply a transitive, symmetric, reflexive, binary relation that partitions a set into equivalence classes. Very simple concept, I'd always assumed this was common knowledge...
If only all of us were as edumacated and intellectshul as you all's conservatives!
I find it hilarious that conservatives spend so much time demonizing smart people ("Intuhlekshul ELEET!"), then turn around and insist that only dumb people would fall for liberalism (which in turn implies that you must be smart, at least by comparison). Can you make up your minds, please? Is being smart good or bad?
Then presumably the test instructions said something like "Solve for what goes in the parenthesis".
Because like I said, if someone asked me what ( ) + 2 was, I'd say 2.
Steve
A work that expires before its copyright never enters the public domain and thus enjoys eternal copyright protection.
This is by far the dumbest article I have ever read. A bunch of middle school kids who can't do math - what do you expect when you give them calculators and Google to do all their thinking for them? When they have to think for themselves, they won't be able to. Just another horrible side effect of technology.