US Students Struggle With Understanding of the 'Equal' Sign 1268
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.'"
Wrong (Score:5, Funny)
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.
It should be: 4+3+2=x+2 (Solve for x) (Score:5, Insightful)
Re:It should be: 4+3+2=x+2 (Solve for x) (Score:4, Funny)
My thoughts exactly.
For me, 4 + 3 + 2 = ( ) + 2
=> 4 + 3 + 2 = (empty value) + 2
=> 4 + 3 + 2 = 0 + 2
=> 9 = 2
=> wtf?
Re:It should be: 4+3+2=x+2 (Solve for x) (Score:4, Funny)
a = b
2. Multiply through by a
a^2 = ab
3. Subtract b^2
a^2 - b^2 = ab - b^2
4. Factor both sides
(a - b)(a + b) = b(a - b)
5. Remove canceling values (a - b)
a + b = b
6. Observing that a = b
b + b = b
7. Combine like terms on the left
2b = b
8. Divide by the non-zero b
2 = 1
???
Re:It should be: 4+3+2=x+2 (Solve for x) (Score:4, Insightful)
Re:It should be: 4+3+2=x+2 (Solve for x) (Score:4, Insightful)
Well, it's sort of that confusing.
For the most part, my math skills are about that of a competent sophomore or maybe junior in high school, which isn't so bad for an adult American these days. I have never seen anyone present an equation like "4 + 3 + 2 = () + 2". To me, that's either a syntax error, or somebody saying "9 = 2", which is just wrong. I've never seen empty parentheses treated as a variable, and I'd be shocked if it's commonly-taught in American schools.
That said, I would never come up with putting 4+3+2 in the parentheses. That's just a WTF.
Re:It should be: 4+3+2=x+2 (Solve for x) (Score:4, Insightful)
It's very confusing. Only after continuing reading the "wrong" solution by students, I realized that he used parentheses for variable names.
FWIW, parentheses usually group statements. In the example there's nothing to group, so I would say that this "non-standard" use is simply wrong.
Re:It should be: 4+3+2=x+2 (Solve for x) (Score:5, Insightful)
I'm pretty sure it's supposed to be an ASCII drawing of a circle, just like [ ] is supposed to be a box, and ( o Y o ) is supposed to be boobs. Lots of primary/middle school textbooks use circles or boxes for the spot you put the answer. So what happens is, a student sees "4 + 3 + 2 = circle" and writes 9 in the circle. They do that all of the time, tests, assignments, that's how it works. You add the numbers and put them in the circle. So, they see "4 + 3 + 2 = circle + 2" and they put 9 in the circle like always, they do the math left to right like they're supposed to, and after they have done that, there is another + 2 after they are done, so they add 9 and 2, since they have "9 + 2" still, now it's 11. It's reasonable, because they were never taught what = means, exactly, just to put the answer in the circle, and to do things left to right. The problem is, they haven't learned algebra yet. So, chuckling about how they couldn't derive it from first principles is just stupid. Show it to them once they've seen algebra. Saying that 70% of americans in grade X got it wrong, but 0% of chinese of the same age were wrong, is meaningless if they teach algebra there sooner. You might say "100% of Chinese who have learned algebra understood algebra, but 'only' 30% of Americans who have never seen algebra, picked it up on the spot".
When I was in grade 9 or 10, I missed like a week of school with a bad case of the flu. I guess we learned algebra that week. When I came back it was test time, and the teacher said I could do it later since I missed the whole section. I said "Naw I'll be fine" and wrote it. I guess I'm in the 30% because after going "Wut" over and over I figured out what it meant. But I can totally see how they could be totally confused by it, circle or x or whatever other placeholder you like. 70% sounds about right for how many wouldn't get algebra if you threw it in their faces with no warning. Obviously, in hindsight everybody on Slashdot would say "OMG SO FUCKING EASY JUST ALGEBRA WHAT RETARDS", but it's not obvious until you have your "ah ha!" moment.
Re:It should be: 4+3+2=x+2 (Solve for x) (Score:5, Insightful)
understanding of equality.
I'm sure they understand equality just fine, it's just that after punching everything into a calculator for all their lives, they don't understand that = means equality instead of "what do the things I just entered equal?"
Re:It should be: 4+3+2=x+2 (Solve for x) (Score:5, Insightful)
Equals always equals equals.
In English - four plus three plus two equals something plus two. That's exactly how I read it, and how everybody I know would read it. Educated in Canada, for clarification.
If anything, the comments on this article really drive home its point. Why are people throwing out the rules when they come upon an unknown? If they understood clearly, concretely, what "equals" meant, there wouldn't be the sort of confusion that's been going on. I think another poster had a good theory, that nowadays "=" is seen as "solve it" due to its use on calculators.
Re:It should be: 4+3+2=x+2 (Solve for x) (Score:5, Insightful)
But just as in a grammar problem you have to choose the right word to put in the blank to make the sentence correct, in a mathematical problem you have to choose a number that results in a correct formula, and that's where they failed. They didn't understand that a formula with an equal sign (an equation) is correct if and only if the two sides have the same value. This is what TFA means by "the meaning of the equal sign".
Null Pointer Exception (Score:5, Funny)
Re:It should be: 4+3+2=x+2 (Solve for x) (Score:5, Insightful)
Exactly.
Are you testing their knowledge of the equal sign? Or are you testing their ability to guess about the meaning of your non-standard notation? This is a common problem that teachers face. I am an ex-teacher. We worked hard (often as teams) to eliminate or rewrite questions like this from our tests and quizzes.
Re:It should be: 4+3+2=x+2 (Solve for x) (Score:5, Insightful)
Why? The world doesn't format problems neatly for you. That's the job of the person approaching it. Simply identifying the mapping to known math formalisms is 90% of the challenge, if not more! If you can't convert a "put more apples on the table and find how many are on it now" into an addition problem, the world won't hold your hand and do it for you.
If the students genuinely understand (or even notice) what they're being taught, they won't be thrown off by stuff like this.
I mean, I'm a little sympathetic, but still, students shouldn't be taught some narrow skill that works *only* for your class. The skills you teach need to be grounded to the rest of the world so they know how it fits in and can adapt to novel situations as necessary.
If their understanding is so brittle that it requires this careful handling before it's a "fair" test, they haven't learned anything, except how to pass tests. Worse, tests presented by *that* teacher.
Re:It should be: 4+3+2=x+2 (Solve for x) (Score:4, Insightful)
Why? The world doesn't format problems neatly for you. That's the job of the person approaching it. Simply identifying the mapping to known math formalisms is 90% of the challenge, if not more! If you can't convert a "put more apples on the table and find how many are on it now" into an addition problem, the world won't hold your hand and do it for you.
OK, how about this one then ...
4#3#2@[]#2
Now I just wrote it and know which arbibrary symbols I replaced the more common ones with, but I still have trouble looking at it and working out what it means! The standardisation of mathematical symbols, and their common use, is what makes it even vaguely teachable. Using "()" as an indicator of a missing term in an equation is madness because everyone I've ever known would use them to indicate a change to the default order of calculation (BODMAS [easymaths.com]). If kids are being taught this way what the hell do the do if they see an equation with braces in it? Ignore the contents and just replace everything in it perhaps?
Re:It should be: 4+3+2=x+2 (Solve for x) (Score:5, Insightful)
The equation noted lacked the precision of mathematics, and is therefore inappropriate without an instruction to the effect of "Solve for the number in () that makes this a true statement."
I'm just an engineer and all, but I had to look at it twice to understand what they were looking for.
Re: Nonstandard notation (Score:5, Insightful)
Re:It should be: 4+3+2=x+2 (Solve for x) (Score:5, Informative)
I saw that in textbooks right on the cusp of learning algebra, to ease you into it. Only, it's not really "( )", it's, well...Slashdot doesn't support unicode so I can't show you...but, it's supposed to be a circle. TFA didn't use one either. IDK if it's because they're not using the same textbooks I've seen, or just because they don't know how to type unicode, either. At any rate, students would, from their earliest years, be used to seeing "2 + 2 = ( )" or "2 + 2 = [ ]" where those are supposed to be circles or boxes for them to put the number in. Or, perhaps "2 + 2 = ___" a blank line for them to put the answer in. The point was that, with no explanation of the equal sign, they come to the wrong conclusion about that circle. They see "4 + 3 + 2 = circle + 2" and they do what they've always done, by rote, and put 9 into the circle, then proceed on to the next little bit, which is +2, there, 9 + 2 is 11, they wonder why there isn't another circle, and make one.
At any rate, your solution of "just use algebra" is absurd, they haven't learned it yet. Algebra is what they're trying to teach them with this. And the point is, it doesn't matter. If they show them "x = 2 + 2, so x is 4", they just might get it. But, if they see "4 + 3 + 2 = x + 2" they would do the same as before "x is 9, so x + 2 is 11". They're just assigning too low a priority to equality in the order of operations, really...and also thinking in C I suppose, where (x = 9) + 2 does equal 11 ;)
Re:Who ever came up with this should be fired. (Score:5, Insightful)
The students are the one who made up the =11 part. Try punching it the question "4+3+2= +2" into a calculator and you'll see why. To the students raised on calculators, "equals" doesn't mean equality anymore, it means "what do the numbers up to here add up to?" So they get to " = ( ) " and perform the "what do the numbers up to here add up to" operation, and write the answer in the blank provided. Then they're left with the +2 bit, so they add it again.
Left to right order of operations, for all operations.
Re:Who ever came up with this should be fired. (Score:4, Insightful)
The problem here is not the use of the equal sign, it is their completely asstarded implementation of the parenthesis that is some how intended to imply one variable twice, with a line break in the middle.
The parenthesis weren’t what triggered that interpretation; the equals sign was. Exactly like a calculator: you calculate, you push “equals”, you get an answer. You calculate some more, you get a new answer.
Re:Wrong (Score:5, Insightful)
That's not what = means. = is ASSIGNMENT. They're looking for ==.
Much as I know you're joking, I'd really love to get rid of this bane that C has brought upon us. Many previous languages used := to mean assignment, hence avoiding the clash with the mathematically well defined = symbol.
Re:Wrong (Score:4, Insightful)
I used various forms of BASIC as a kid, and = was fine there. I had some formal education in Pascal, = was fine there.
Now when I occasionally do a little scripting in a modern language, I spend most of my time tearing my hair out at bugs which turn out to be the result of me using = when I should have put ==
I'm sure there are good reasons for it that make sense to proper programmers, but personally I'd like to give whoever came up with this syntax a kick in the bollocks. Why would I want to do an assignment in an if statement or a loop condition check anyway?
Re:Wrong (Score:4, Funny)
For those who don't know, the guide to school in America and England:
America - England
Math - Maths
Science - Sciences
Art - Arts
Gym - Gyms
Lunch - Lunches
Recess - Recesses
Detention - Detentions
Re:Wrong (Score:4, Informative)
Re:Wrong (Score:4, Informative)
Re:Wrong (Score:5, Funny)
In most of the world we study Mathematics. I didn't realise that there was only one Mathematic studied in the US.
Is it really plural, though? (Score:5, Funny)
Do your Barbie dolls say "Maths are hard"?
Re:Is it really plural, though? (Score:5, Informative)
Re:Is it really plural, though? (Score:5, Informative)
Sice everybody seems to be too lazy to check, here's an excerpt from Wikipedia.
Etymology
The word "mathematics" comes from the Greek (máthma), which means learning, study, science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times.[9] Its adjective is (mathmatikós), related to learning, or studious, which likewise further came to mean mathematical. In particular, (mathmatik tékhn), Latin: ars mathematica, meant the mathematical art.
The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural (ta mathmatiká), used by Aristotle, and meaning roughly "all things mathematical"; although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from the Greek.[10] In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in English-speaking North America, math.
Re:Wrong (Score:5, Insightful)
Re:Wrong (Score:5, Funny)
Reading your post, I got to "ematic" and thought it was some language term for a word fragment that I didn't know, until I realized it WAS the word fragment.
Slashdot reading in the early morning is hard.
Re:Wrong (Score:5, Insightful)
Math sounds awkward to me, because I was brought up with Maths. This is like an essay I read ages ago about why rear wheel drive is more natural than front. I thought it was a load of crap because I'd learned to drive in FWD vehicles and my natural driving instincts in certain situations were different to what the guy said that the "natural" was.
For most things in life whatever is more "natural" for you often depends on what you were brought up with/trained on.
Re:Wrong (Score:5, Informative)
In most of the world we study Mathematics. I didn't realise that there was only one Mathematic studied in the US.
This is a dialectical thing about American English. We use singular verb inflections with collective nouns.
Queen's English: "Aerosmith are playing Wembley Stadium."
U.S. English: "Aerosmith is playing the Verizon Center."
This is why you hear "the data is" over here.
Re: (Score:3, Insightful)
And how about Economics, Politics, Aeronautics, and Quantum Mechanics?
Re:Wrong (Score:5, Funny)
Re:Wrong (Score:5, Insightful)
Re:Wrong (Score:4, Informative)
Re:Wrong (Score:5, Insightful)
Re:Wrong (Score:5, Funny)
Does it matter that England, Scotland, Wales and Northern Ireland are different countries and are referred to as the United Kingdom of Great Britain and Northern Ireland (the inhabitants being commonly referred to as British)?
I believe it is well established that people from the USA are American and those from the UK are British - can we leave it at that?
Whilst we are at it, when Americans hear what they perceive to be an Australian accent, assume that the person speaking is in fact from the UK. Most Australians are drunk and wear hats with corks hanging from strings.
In return the British will listen out for an American accent with a strange twang and automatically assume the person speaking is from Canada. We will also look out for the hockey stick and racoon skin hat but will be wary of the hat without the stick as that person is likely to be Russian.
What's wrong with Ohioan and Wyomingite? (Score:4, Funny)
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 ;)
Re:Wrong (Score:5, Funny)
Or pasty white crumpet monkeys.
Re:Wrong (Score:4, Insightful)
BASIC is perfectly sane. There are clean, contextual rules which disambiguate between = the assignment operator and = the equality test.
Let's take a moment to remember that "x = 1" is only a legal BASIC statement in the first place because interpreters have been relaxed for programmers too lazy to use "Let".
Re:Wrong (Score:4, Informative)
"x = 1" is only a legal BASIC statement in the first place because interpreters have been relaxed for programmers too lazy to use "Let".
It's not really laziness. Mostly it's for historical reasons. My first computer (a TRS-80 model 1, "Level 1") had 4K of RAM. That's right, 4096 bytes. When you've got that little space for your BASIC programs, removing the requirement for "let" in your BASIC code freed up valuable bytes. (other similar shortcuts existed back in the day, such as "?" available as a replacement for "print".) ...and it wasn't just TRS-80s. Most computers of that era had similar restrictions. Today, space is virtually unlimited, but these legacy shortcuts remain.
Now get off my lawn.
Re:Wrong (Score:4, Insightful)
Back then (80s) programming was the only way to use the TRS-80s and Apple IIs the schools gave us. Today? You just need to learn how to turn them on and click an icon, and so programming is no longer considered necessary unless you're going into a CSE major.
BTW:
I see a problem with the problem in the summary: 4+3+2=( )+2 is not the way math questions are typically phrased. In my experiences these problems usually looked like this: "4+3+2 = __+2 ; Fill in the blank." The instructions were explicit so students did not need to guess the teacher's desired result.
I don't like teachers that think writing confusing tests (aka trick questions) is any test of student ability. It's more a demonstration of the test-writer's lack of communication skills.
Well, that explains things. (Score:4, Insightful)
So I'm not being a curmudgeonly old jackass when I think this generation is stupid.
I guess I'm stupid, too. (Score:5, Funny)
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?
Re:I guess I'm stupid, too. (Score:5, Informative)
Re:I guess I'm stupid, too. (Score:4, Insightful)
Thats what I gathered too, and it was a bit confusing to read. Knowing parenthesis as delimiters for so long, it was strange to see. I wonder if that is what they showed to the kids, and how it would have been different if they used something like:
4 + 3 + 2 = ? + 2
Re:I guess I'm stupid, too. (Score:5, Informative)
The actual notation used in math questions and textbooks is a blank space (e.g., an underlined blank space). The parenthesis are a poor attempt and rendering that in text.
Re:I guess I'm stupid, too. (Score:5, Insightful)
___, this is very ____ to do.
Re:I guess I'm stupid, too. (Score:4, Insightful)
___, this is very ____ to do.
I can find at least two solutions :
Re:I guess I'm stupid, too. (Score:4, Funny)
Re:I guess I'm stupid, too. (Score:5, Informative)
I had to read it twice to get what they wanted done. An empty set of parens in proper mathematical expressions is valid and equivalent to (0).
"4+3+2=x+2 solve for x" is the correct way to state that problem.
Re:Well, that explains things. (Score:5, Interesting)
Re:Well, that explains things. (Score:5, Funny)
Re:Well, that explains things. (Score:4, Informative)
Re:Well, that explains things. (Score:5, Funny)
Now, now. Just because these youngsters need pictures of the food on their cash-register buttons in order to do their job doesn't mean they're stupid. :-)
Re:Pictures (Score:5, Insightful)
So that explains the MS Office Ribbon?
Re:Well, that explains things. (Score:5, Insightful)
You're absolutely correct, it doesn't. And, in fact, they don't need them at all. Comments like this just show your ignorance of how organizations work at large scale. The pictures are there because they are universal.
And those are just the three reasons that are most obvious to me. Now get off your high horse!
Re:Well, that explains things. (Score:5, Insightful)
So I'm not being a curmudgeonly old jackass when I think this generation is stupid.
I think there's still a chance you are. Is it not more likely that rather than this generation being stupid, it is just being taught poorly by your generation? The article talks about the method students use to solve an equation. Why would a whole generation of students use a different method (and the same method) than the previous generation unless they were taught that method.
Re:Well, that explains things. (Score:4, Insightful)
It's a sine of the times.
Re:Well, that explains things. (Score:5, Insightful)
So I'm not being a curmudgeonly old jackass when I think this generation is stupid.
Oh hell. Generation N has always claimed that Generation N+1 is {stupid, lazy, amoral, immoral, bound for *insert cultural analogy to Hell*}. This holds inductively for all values of N. Strangely enough, they also happen to think that Generation N+2 is cute and cuddly.
I hate to tell you, but our parents' generation thought we were idiots too, I'm sure. I know their parents thought they were.
Re:Well, that explains things. (Score:5, Insightful)
Re:Well, that explains things. (Score:5, Insightful)
While it's true that just because they thought it doesn't mean it's true, the opposite is likewise valid: just because many previous generations thought it, does not make it false.
It provides a burden for uniqueness that needs to be met, though.
It used to be that wisdom of age was respected and revered, even taken to heart.
When? When in living memory did the majority of young people actually respect their elders? Again, you're repeating things that have been said since the beginning of time. Hell, I've seen almost the exact same thing written in the bible. Kids were assholes then too. They still are. Time goes on.
We're talking about basic, first grade mathematics concepts here. How is this "not getting stupid"?
No, you're actually talking about pre-algebra if you look more closely at the example. Which kids have always generally sucked at.
The last couple generations, however, have been increasingly "stupid" in the "can't solve for x" sense. Test scores clearly prove this.
If there's a problem, it's not with the gene pool of the kids or their abilities. It's caused by well-meaning but catastrophically stupid policies that prevent the removal of problem students from classes, and the elimination of ability-based tracking. This means that normal kids are surrounded by juvenile delinquents and children who don't even speak English. If you remove those students (who would have not taken the test in prior generations) from the scoring, I wonder how the stats would play out.
In other words - it's not that the kids are getting stupid. It's that our schools are completely failing them.
teachers (Score:4, Insightful)
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.
Re: (Score:3, Insightful)
Calculators (Score:5, Insightful)
I blame it on calculators where the evaluate button has "=" on it.
I don't understand the example, either (Score:3, Insightful)
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.
Confusing symbols (Score:5, Interesting)
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?
Re:Confusing symbols (Score:5, Informative)
No, we use 'x' over here, too.
Re:Confusing symbols (Score:5, Informative)
we used BOTH an x and a 'box' (as per my other post).
starting out, they taught us to fill in the missing value in the 'box' (square symbol). then, over time, when it was the right time to introduce letters as 'box symbols' they put an 'x' there.
made sense to me. a progression to get the kid up to that level of thinking. a box is empty and can be filled. makes good concrete sense. then later, we 'upgrade' the box to an x. same concept but more steps to get the kid there.
Re:Confusing symbols (Score:4, Insightful)
Yes, and no doubt the teacher worked a few problems on the board so that everyone could see how they were done.
And since everyone daydreamed through the class, the homework got done with the calculator.
4
+
3
+ (calculator displays 7)
2
= (calculator displays 9; write it down)
+
2
Now to finish the problem? Well, = of course, and write down “= 11”. That’s what the calculator said.
Re:Confusing symbols (Score:5, Interesting)
I'm from the UK, is 4+3+2=( )+2 a commonly used / commonly understood way of presenting the problem in the US?
It sure isn't. I wonder if notational trickery isn't part of the problem, not a lack of understanding. (TFA doesn't say if there were directions, like "Solve for the missing quantity in parentheses" or something like that.) I bet more people would have understood if they used something like x. Maybe they were trying to avoid "scary" variables for middle schoolers, but that's actually exactly when I remember learning what they were--if not, the year before.
Re:Confusing symbols (Score:5, Informative)
If you watch the video, they have pictures of the math questions, which makes things a lot clearer. The parentheses are TFA's way of trying to draw a blank space. In the original questions, it's an underlined blank space (so ___ would have been a better choice) -- the same sort of underlined blank space provided in grade school where they want you to fill in the answer. In mathematics classes before algebra, when they're trying to introduce you to algebraic concepts, it's common to use blank spaces for "figure out what goes in this space and write it", rather than writing an "x" and saying "solve for x", which would use a concept the students haven't yet been taught.
This is GREAT NEWS (Score:5, Insightful)
It means that even after China abolishes it's sweatshops there will still be a source of cheap unskilled labor in the world.
4+3+2=( )+2 (Score:5, Funny)
Let's see here.. I'm going to go with:
4+3+2=(21/3*981727612785316256514034236^0)+2
Don't know what () means (Score:4, Insightful)
Re: (Score:3, Insightful)
Ah. So () is a helmet. Kids wear helmets for everything these days.
Is that really the best example (Score:3, Insightful)
Petkovsek, Wilf, Zeilberger A=B (Score:3, Interesting)
For an expanded explanation of what the equals sign means, check out Petkovsek, Wilf and Zeilberger's A=B [rutgers.edu]. I remember it as a very enjoyable read from university, in parallel with Concrete Mathematics [wikipedia.org]... (btw, why won't š show in comments?)
The problem (Score:4, Insightful)
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.
Understanding and not memorization is the key (Score:4, Interesting)
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.
Math education in America is pathetic (Score:5, Interesting)
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.
Re: (Score:3, Informative)
Its because you're shoving them into one equation. The scope of your 'working out' is to solve whatever is in that equation so the correct answer to
1+2 = 3
is
3 = 3
true.
RTFA, it's not that usage which he's objecting to (Score:5, Insightful)
"'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.
Re:RTFA, it's not that usage which he's objecting (Score:4, Interesting)
I also wonder how these kids cope when a second variable is introduced.
Re:RTFA, it's not that usage which he's objecting (Score:4, Interesting)
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.
Re: (Score:3, Insightful)
To put it another way, he's saying that the students are treating mathematical expressions as a list of instructions to be obeyed, and not as expressions. This works fine for 1+2=? or 4/3=?, but leads to a cognative train wreck when trying to deal with even the simplest algebra. A student who works that way could never figure out what length of crossbeam they'd need to brace a 3x4 wooden frame.
Re: (Score:3, Informative)
Ya, I've never understood why 1 + 2 = 3 = 4 - 1 isn't okay.
Technically it is as okay as it gets, both sides of each equality operator is equal which is exactly how the symbol works. TFA is about how people don't actually "get" that, if you look at the example in the summary it essentially says 9 = 11 which of course is plain wrong.
The reason "double equalities" might be wrong is if you're solving an equation while showing each step.
Re: (Score:3, Funny)
Additionally, as you can see, our President wants to KILL SMURFS!
Re:Home School (Score:5, Insightful)
This is one reason why we home school...public school systems fail in so many ways.
A better solution is to find a better school. A better public school, or a private school, or a charter school, or something.
Yes, home schooling can be used to impart better information. You've got a much smaller class size. You've got more attachment to your pupil. You can devote as much time and effort to educating your kid as you feel necessary.
But home schooling pretty much fails to develop a kid's social skills. And I've always felt that one of the more important things that public schooling does is develop social skills.
Home schooled kids don't generally have to put up with schoolyard bullies. They don't have to make friends. They don't learn about compromises and sharing and common interests the way you do when you're surrounded by other people all day long. They don't learn to file the rough edges off their own personality, so that they can get along with others. They don't learn how to put up with other people's quirks and issues. They don't learn diplomacy and tact.
Yes, you can supplement your home schooling with some good social exposures... Send your kid to the park for a good chunk of the day, or get them involved in some kind of sports or clubs... But, from what I've seen, an awful lot of folks who are doing home schooling aren't interested in exposing their kids to much of anything. They're more concerned about sheltering their kid either from harm, or from opposing viewpoints.
Some of the hardest people I've had to work with are those who've been home schooled. They're generally very smart, very well-educated, and completely unable to deal with other human beings.
Re:Home School (Score:4, Insightful)
But home schooling pretty much fails to develop a kid's social skills. And I've always felt that one of the more important things that public schooling does is develop social skills.
Oh God, I am so tired of hearing this BULLSHIT. What social skills are you talking about? Let's take the U.S. for example. Most people here have been through the public school system, correct? Yet by just about any metric, people here are a bunch of selfish assholes. Look at the divorce rate. Look at the mudslinging on any general public forum like Yahoo message boards. Look at the way people behave on the highway or on Black Friday. People's social skills unilaterally SUCK. I don't believe that homeschooling is going to produce a less socially adapted adult. Really. It's bullshit.
Re:Home School (Score:5, Interesting)
I'm going to guess you're home-schooled, both because of your defensiveness and because of your lack of analytical skills (as opposed to memorization).
And you would be incorrect. Which means that I am a social paragon. Actually, just put me under the column of counter-example.
They made a comparison between two subject groups, which you did not address.
They didn't address it either except with some anecdotal evidence. Most of the socially maladjusted people I have met are products of the public school system. So, there is my anecdotal evidence. It cancels out. Care to bring any actual research to the discussion?
In fact, the existence of bullies undermines the argument, since those folks are by definition not socially well-adapted, and furthermore, they tend not to be "cured" by graduation.
In fact, let's roll with their thesis. Fire all the teachers, principals, etc. and just send the kids into a locked building with no adults for 6 hours a day. Of course, we shall expect them to not only master the social graces, but be experts in automobile engineering, medicine, mathematics, literature, composition, business, and pretty much everything else. Why not? They can learn all that stuff on their own, just like they can learn social skills in an unsupervised setting.
Who is lacking the analytical skills? Really?
Re:Home School (Score:5, Insightful)
Are you nuts?
"Home schooled kids don't generally have to put up with schoolyard bullies. They don't have to make friends. They don't learn about compromises and sharing and common interests the way you do when you're surrounded by other people all day long."
All the homeschooled kids around where I grew up dealt with all that.
The asshole kids that bullied also bullied the home-schooled kids, granted they did not get to deal with the imported bullies from across town, but a bully is a bully. and they made friends with kids that lived near them.. Plus many were in sports programs with the public school kids. You can be home schooled and play football for your local public school at the same time. They joined lots of extra curricular activities. Many of us were jealous as they typically had a 4 hour school day plus got to take "classes" we never got. One kid was taking a class at the local motorcycle shop for learning small engine repair at 13 years old.
I know you guys love your twisted view of homeschooled kids as all living in basements and named "wolfgang" or "moon-unit-alpha" and are never let outside... but it's not reality. in fact it's pretty darn close to racism in being flat out wrong.
Re:Home School (Score:4, Insightful)
Are you nuts?
Not as far as I know...
"Home schooled kids don't generally have to put up with schoolyard bullies. They don't have to make friends. They don't learn about compromises and sharing and common interests the way you do when you're surrounded by other people all day long."
All the homeschooled kids around where I grew up dealt with all that.
The asshole kids that bullied also bullied the home-schooled kids, granted they did not get to deal with the imported bullies from across town, but a bully is a bully. and they made friends with kids that lived near them..
So, you're saying that my anecdotal experience is not the same as your anecdotal experience?
Plus many were in sports programs with the public school kids. You can be home schooled and play football for your local public school at the same time. They joined lots of extra curricular activities. Many of us were jealous as they typically had a 4 hour school day plus got to take "classes" we never got. One kid was taking a class at the local motorcycle shop for learning small engine repair at 13 years old.
You did read the full text of my post, didn't you?
Specifically, the bit where I said:
Yes, you can supplement your home schooling with some good social exposures... Send your kid to the park for a good chunk of the day, or get them involved in some kind of sports or clubs... But, from what I've seen, an awful lot of folks who are doing home schooling aren't interested in exposing their kids to much of anything. They're more concerned about sheltering their kid either from harm, or from opposing viewpoints.
Like it or not, the folks who were home schooled when I was growing up did not turn out to be well-rounded individuals.
Like it or not, most of the folks that I've found very difficult to work with have turned out to be home schooled.
And since I'm not omniscient, I can only speak from my own relatively small chunk of life experiences.
Re:Home School (Score:4, Insightful)
That's because the homeschooled adults with bad social skills are the ones he notices; he's never realized that the homeschooled adults with good social skills exist, because it never occurs to him to ask adults with good social skills how they were schooled.
Re:Home School (Score:4, Interesting)
If you could get rid of all the religious wackos who use homeschooling as a way to indoctrinate their kids without having them exposed to the outside world, there would be a lot less stigma on the homeschooled crowd.
Re:Home School (Score:5, Interesting)
[citation needed]
Here I am. Cite me: I'm an original source.
My kids go to the public Montessori school in my city. It only goes to 3rd grade, though, and after that they have to transfer. We'd heard good things about another local public school and enrolled my daughter there. It was a fucking disaster.
Examples:
In 3rd grade, she was doing square roots. Her new 4th grade class was starting to learn 2-column addition.
The class had one hour to do math problems 1 through 20 (out of 100). She finished in about 10 minutes because it was remedial to her. For want of something to do, she went ahead and finished problems 21-100. Her teacher called her out in front of the entire class: "The assignment was 1-20, not 1-100. You didn't pay attention."
Since she wasn't allowed to work ahead, she pulled out a book to read. Again, from the teacher: "This is math, not reading. Put that away!" She was literally required to sit quietly in her desk for the remaining time.
Her weekly list of words to memorize for the spelling test on Fridays included "off" and "zoo". In 4th grade. I swear to God that I'm not exaggerating.
She'd cry in the mornings. "Please don't make me go to school today! I hate it there! Can't you tell them I'm sick and work from home and let me stay here with you?" I'd be sad if she was saying those things 6 years from now. Coming from my 4th grader, it broke my heart.
So I went to talk to the principal. He was a nice guy, and I'm a nice guy. We had a great visit and he said he'd work with the teacher to find more challenging work for my daughter to keep her busy and interested. We shook hands and I left.
Within the week, my daughter got detention for "looking bored in class". Shortly after that, she got a 96% on a test. Her teacher asked her (yet again, in front of the class) why she doesn't "get perfect scores on all her tests if [she's] so smart."
That afternoon, I enrolled her in a different local private school. They were doing cubes and long division in math class, and learning Latin and Greek word components in language class.
Sometimes it's not enough to talk to the teachers and administration.
Re:Home School (Score:4, Insightful)
Re:Headline should read... (Score:4, Insightful)