typodupeerror

## US Students Struggle With Understanding of the 'Equal' Sign1268

Posted by Soulskill
from the confusing-parallel-lines dept.
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.'"
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## US Students Struggle With Understanding of the 'Equal' Sign

• #### Re:Wrong (Score:1, Interesting)

on Friday August 13, 2010 @09:36AM (#33238268) Homepage Journal

I wonder how much this has to do with programming, and the fundamentally different nature of the meaning of = in that and maths? Yeah, you heard me Americans, maths plural.

• #### Confusing symbols (Score:5, Interesting)

on Friday August 13, 2010 @09:39AM (#33238330)
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?
• #### Re:Well, that explains things. (Score:5, Interesting)

on Friday August 13, 2010 @09:42AM (#33238378)
I have a friend who's a high school teacher. He's been predicting the downfall of society for a few years now, based on the fact the kids he teaches are - for the most part - useless twats. What makes it even worse is they also carry a strong sense of entitlement, as in "even though I can't be bothered to do the work properly or learn a single fucking thing while I'm here, I deserve an A grade from you, and when I graduate I am going to deserve an \$80K starting salary somewhere just for showing up and playing FarmVille all day."
• #### Re:Confusing symbols (Score:5, Interesting)

on Friday August 13, 2010 @09:52AM (#33238560)

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.

• #### Petkovsek, Wilf, Zeilberger A=B (Score:3, Interesting)

on Friday August 13, 2010 @09:53AM (#33238576)

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 &scaron; show in comments?)

• #### Re:Wrong (Score:3, Interesting)

on Friday August 13, 2010 @10:03AM (#33238780)

I wonder how much this has to do with programming, and the fundamentally different nature of the meaning of = in that and maths?

There's no fundamental meaning of = in programming... There's a meaning that C gave it, which happens to disagree with what most definitions of the symbol mean. Life would have been so much simpler if C had stuck with the previously used := for assignment.

• #### What? (Score:3, Interesting)

on Friday August 13, 2010 @10:07AM (#33238864) Journal

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.

• #### Re:RTFA, it's not that usage which he's objecting (Score:4, Interesting)

on Friday August 13, 2010 @10:11AM (#33238938) Journal

I also wonder how these kids cope when a second variable is introduced.

• #### Re:Calculators in school (Score:3, Interesting)

<biryokumaru@gmail.com> on Friday August 13, 2010 @10:18AM (#33239056)
My electrical engineering professors seemed to be of the opinion that we were allowed to use a computer when we knew how to design it from scratch.
• #### Re:Wrong (Score:2, Interesting)

on Friday August 13, 2010 @10:22AM (#33239132) Homepage
I am so SICK of that stereotype. Having been all over Europe in the last two years, there are just as many overweight people there as in the US. Possibly more on a per capita basis from what I saw. And Scotland? Holy crap!
• #### Re:Home School (Score:2, Interesting)

on Friday August 13, 2010 @10:29AM (#33239328)
I feel vouchers will address many issues with the public school system. I vote for people who support putting in a voucher system. Other than voting my energy will primarily be focused on my children's upbringing and not "the public" upbringing. My wife is the alpha educator in our house.

We can't agree on healthcare or retirement policies how on Earth would I convince others to adopt a specific learning methodology?

For those interested a great read on our public education system: http://www.amazon.com/Weapons-Mass-Instruction-Schoolteachers-Compulsory/dp/0865716315 [amazon.com]
• #### Re:Wrong (Score:3, Interesting)

on Friday August 13, 2010 @10:31AM (#33239386)
In third grade I wrote a BASIC program that rendered a flying saucer landing. The assignment was to draw a static picture with pixel output. That experience probably guided me to my current profession. I didn't own a computer until the 90's so I would not have gotten that experience any other way.

It would be a shame if they weren't still exposing children to programming in school.

Of course, I did get a similar "how do things work" experience by disassembling Omega Supreme and a number of other toys. :)
• #### Understanding and not memorization is the key (Score:4, Interesting)

on Friday August 13, 2010 @10:33AM (#33239422)

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.

• #### Re:Home School (Score:5, Interesting)

by Anonymous Coward on Friday August 13, 2010 @10:35AM (#33239492)

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:Wrong (Score:3, Interesting)

on Friday August 13, 2010 @10:42AM (#33239704)

In a sane language, '=' would not be used for assignment.

• #### Re:RTFA, it's not that usage which he's objecting (Score:4, Interesting)

<foobarsoft@foobarsoft.com> on Friday August 13, 2010 @10:56AM (#33240000) Homepage

4+3+2 is not equal to 9+2.

That's the problem the students have. My reading has it going like this:

• 4+3+2=()+2
• 9=()+2
• OK, the answer to the first part was 9, so put that in the blank in part B
• (9)+2
• Now I can get an answer
• 9+2=11

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.

• #### And I have to call BS on the writers of the study. (Score:3, Interesting)

on Friday August 13, 2010 @10:58AM (#33240060)

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?

• #### Re:Wrong (Score:2, Interesting)

on Friday August 13, 2010 @11:12AM (#33240350) Journal

Actually, originally in BASIC you had to write: "LET a=5" to set a to 5. However, BASIC implementations tried to reduce typing and allowed omitting the LET (I wonder how many people never new that LET actually existed). There were also other abbreviations like "?" for "PRINT" (that one always puzzled me: How could one get the idea to encode "PRINT" as "?", which is a sign which actually implies asking for something? The more logical way would have been to shorten "INPUT" to "?", and then maybe shorten "PRINT" to "!").

• #### Math education in America is pathetic (Score:5, Interesting)

on Friday August 13, 2010 @11:51AM (#33241100) Homepage

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:Wrong (Score:2, Interesting)

on Friday August 13, 2010 @12:00PM (#33241260)
Give me a break. It's nice to shout at the 80s and everything, but 90% of what C was used then for has being replaced by a language which does treat that as an error - Java. For the other 10%, modern C++ compilers will surely give a warning in this case.
• #### Re:Home School (Score:5, Interesting)

<kirk+slashdot@strauser.com> on Friday August 13, 2010 @12:02PM (#33241306) Homepage Journal

[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:Wrong (Score:3, Interesting)

on Friday August 13, 2010 @12:21PM (#33241622)

That's a bit of a strong statement.

I don't want to pretend I'm some godlike C programmer, but competent coding and a simple review process catches this stuff pretty easily. And sometimes that there if(a=b) is exactly what you mean.

Can't think of any situations off hand, but I'm sure there have been some.

• #### Re:It should be: 4+3+2=x+2 (Solve for x) (Score:3, Interesting)

on Friday August 13, 2010 @01:09PM (#33242352)

So, it's a textbook problem in your opinion. I think that you're right (and that kids shouldn't be using calculators for quite a while into math).

My daughter's math textbook is more old-fashioned, and it has find the missing addend type questions starting in 2nd grade IIRC. They use the notation of an empty box for the child to fill in.

• #### Re:It should be: 4+3+2=x+2 (Solve for x) (Score:3, Interesting)

on Friday August 13, 2010 @01:55PM (#33243142) Journal
I'm not sure if the brackets were literal or just a typographical convention. However historically it wasn't uncommon to see boxes instead of variables at the younger grades - they may still do today. Also without the paper it's difficult to say if this is the only kind of test they put to students. I think your argument is kind of bogus though - You could just as easily level the same criticism at word problems...or perhaps you believe that word problem need a highly rigorous and pre-defined format. Math, as I see it anyway isn't just taught so that you can manipulate symbols deterministically. To me a good test of *knowing* math is to be able to recognize and solve mathematical problems without needing it to be spoon fed to you. Otherwise...what's the point? Machines already outstrip our ability to compute - why outside of extremely simple addition and subtraction would you bother ever learning a lot of math EXCEPT to use it so that you can recognize solve problems that are not handed to you in exam question format.
• #### Re:Home School (Score:4, Interesting)

<slashdotNO@SPAMpitabred.dyndns.org> on Friday August 13, 2010 @02:04PM (#33243292) Homepage

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: Nonstandard notation (Score:2, Interesting)

on Friday August 13, 2010 @02:26PM (#33243676) Homepage
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.
• #### Re:It should be: 4+3+2=x+2 (Solve for x) (Score:2, Interesting)

on Friday August 13, 2010 @04:11PM (#33245148)

Yes, my 3rd grader is given that kind of problem, but with a longish underscore instead of the parentheses. Sometimes it's a box, but the underscore is better, because it's a familiar holdover from learning to read, where a drawing of a feline is accompanied by "__at" and they're told to fill in the blank. I agree that using parentheses would be as poor a choice as "()at" when you want them to produce the three letter synonym for feline.

The mathematical pedagogy is fine; they're trying to develop numeracy (numeric literacy) by instilling the idea that there's more than one way to "express" 7, but they don't (yet) want to muddy the waters more by mixing letters and numbers and using phrases like "solve for"

• #### Re:Wrong (Score:3, Interesting)

on Saturday August 14, 2010 @01:00AM (#33248946) Journal

The story goes that K&R did a statistical analysis of some existing code, and found that assignment was 30% more frequent than comparison, so they went with the shorter version for assignment. But := and = are more historically correct in that they match mathematical operators.

I kinda like the assignment in ML language family, though:

x <- 456;

• #### Re:Wrong (Score:3, Interesting)

on Saturday August 14, 2010 @02:25PM (#33251902)

Hey, I thought about my comment and you're right - there's really no reason not to use the automated tools that are available. It is manifestly possible to produce rock solid, enterprise grade software without them, but from an efficiency perspective you're spot on.

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