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US Students Struggle With Understanding of the 'Equal' Sign 1268

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

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  • Wrong (Score:5, Funny)

    by dyingtolive (1393037) <brad.arnett@NOspam.notforhire.org> on Friday August 13, 2010 @09:35AM (#33238242)
    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.
    • by Knertified (756718) on Friday August 13, 2010 @09:50AM (#33238530)
      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.
      • by indeterminator (1829904) on Friday August 13, 2010 @10:07AM (#33238866)

        My thoughts exactly.

        For me, 4 + 3 + 2 = ( ) + 2
        => 4 + 3 + 2 = (empty value) + 2
        => 4 + 3 + 2 = 0 + 2
        => 9 = 2
        => wtf?

      • by SpeZek (970136) on Friday August 13, 2010 @10:08AM (#33238886) Journal
        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.
        • by NNKK (218503) <nknight@runawaynet.com> on Friday August 13, 2010 @10:39AM (#33239608) Homepage

          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.

        • by he-sk (103163) on Friday August 13, 2010 @10:41AM (#33239658)

          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.

          • by canajin56 (660655) on Friday August 13, 2010 @12:20PM (#33241600)

            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.

        • by Qzukk (229616) on Friday August 13, 2010 @10:57AM (#33240026) Journal

          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?"

      • by Senior Frac (110715) on Friday August 13, 2010 @10:11AM (#33238950) Homepage

        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.

        • by DriedClexler (814907) on Friday August 13, 2010 @10:40AM (#33239634)

          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.

          • by AGMW (594303) on Friday August 13, 2010 @11:29AM (#33240698) Homepage

            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?

          • by aaarrrgggh (9205) on Friday August 13, 2010 @11:40AM (#33240882)

            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.

      • by Sithech (858269) on Friday August 13, 2010 @10:20AM (#33239088)
        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.
      • by canajin56 (660655) on Friday August 13, 2010 @12:04PM (#33241362)

        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:Wrong (Score:5, Insightful)

      by beelsebob (529313) on Friday August 13, 2010 @10:01AM (#33238762)

      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)

        by mike2R (721965) on Friday August 13, 2010 @10:22AM (#33239118)
        Ok I'm going to display my ignorance here and ask why isn't = on its own good enough for a comparison?

        I used various forms of BASIC as a kid, and = was fine there. I had some formal education in Pascal, = was fine there.

        Now when I occasionally do a little scripting in a modern language, I spend most of my time tearing my hair out at bugs which turn out to be the result of me using = when I should have put == .

        I'm sure there are good reasons for it that make sense to proper programmers, but personally I'd like to give whoever came up with this syntax a kick in the bollocks. Why would I want to do an assignment in an if statement or a loop condition check anyway?
  • by dr_strang (32799) on Friday August 13, 2010 @09:35AM (#33238246)

    So I'm not being a curmudgeonly old jackass when I think this generation is stupid.

    • by maillemaker (924053) on Friday August 13, 2010 @09:41AM (#33238352)

      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?

    • by oldspewey (1303305) 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."
    • by Zeek40 (1017978) on Friday August 13, 2010 @09:43AM (#33238384)
      Not just stupid, defiantly and proudly stupid. We've devolved into a culture that celebrates its own ignorance
    • by fahrbot-bot (874524) on Friday August 13, 2010 @09:44AM (#33238402)

      So I'm not being a curmudgeonly old jackass when I think this generation is stupid.

      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)

        by TaoPhoenix (980487) <TaoPhoenix@yahoo.com> on Friday August 13, 2010 @09:58AM (#33238688) Journal

        So that explains the MS Office Ribbon?

      • by Proteus (1926) on Friday August 13, 2010 @03:17PM (#33244390) Homepage Journal

        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. :-)

        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.

        1. McDonald's (for example) is an international company, and they serve their core menu in dozens of languages. It's much easier and less error-prone for them to produce a picture-based keypad than to translate everything without error
        2. Fast food companies did research indicating that it's faster -- even for highly-literate people -- to find an item by image rather than by name. Faster means better service with fewer staff, which means more profit.
        3. Many fast food chains, and McDonald's in particular, hire people with disabilities. This is a huge win for such people -- real, productive work that can help make them at least partly independent. Many with cognitive or developmental disabilities have written-language challenges, and the picture "menus" are much easier for them to use efficiently. It doesn't make sense to have two versions of something if one works for everyone, does it?

        And those are just the three reasons that are most obvious to me. Now get off your high horse!

    • by KnownIssues (1612961) on Friday August 13, 2010 @09:46AM (#33238464)

      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.

    • by theskipper (461997) on Friday August 13, 2010 @10:00AM (#33238724)

      It's a sine of the times.

    • by Mr. Underbridge (666784) on Friday August 13, 2010 @10:03AM (#33238784)

      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.

      • by TapeCutter (624760) * on Friday August 13, 2010 @11:43AM (#33240974) Journal
        "Our youth now love luxury. They have bad manners, contempt for authority, they show disrespect for their elders and love chatter in place of exercise; they no longer rise when elders enter the room; they contradict their parents, chatter before company, gobble up their food and tyrannize their teachers." - Socrates, (circa 400BC)
  • teachers (Score:4, Insightful)

    by flynt (248848) on Friday August 13, 2010 @09:38AM (#33238284)

    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)

      by NEDHead (1651195)
      Since I doubt you can produce someone older than 500 years, so far as everyone is concerned the = symbol has been around forever. If anyone beyond 3rd grade cannot understand the problem and solve that equation for the unknown value placeheld by the ( ) symbol then the teachers' unions should take the blame.
  • Calculators (Score:5, Insightful)

    by Anonymous Coward on Friday August 13, 2010 @09:38AM (#33238296)

    I blame it on calculators where the evaluate button has "=" on it.

  • by petes_PoV (912422) on Friday August 13, 2010 @09:39AM (#33238316)
    Are they saying the quantity denoted by the braces () is an unknown and we should solve for it. Are they saying it's some sort of sub-total of evaluating the LHS of the expression? from the rest of the text:

    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)

    by M_Hulot (859406) 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:Confusing symbols (Score:5, Informative)

      by flynt (248848) on Friday August 13, 2010 @09:45AM (#33238422)

      No, we use 'x' over here, too.

      • Re:Confusing symbols (Score:5, Informative)

        by TheGratefulNet (143330) on Friday August 13, 2010 @09:56AM (#33238644)

        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.

        • by clone53421 (1310749) on Friday August 13, 2010 @10:39AM (#33239600) Journal

          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)

      by R.Mo_Robert (737913) 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.

    • Re:Confusing symbols (Score:5, Informative)

      by blueg3 (192743) on Friday August 13, 2010 @09:57AM (#33238668)

      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)

    by lxs (131946) on Friday August 13, 2010 @09:41AM (#33238356)

    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)

    by batquux (323697) on Friday August 13, 2010 @09:42AM (#33238370)

    Let's see here.. I'm going to go with:
    4+3+2=(21/3*981727612785316256514034236^0)+2

  • by Neil Watson (60859) on Friday August 13, 2010 @09:42AM (#33238376) Homepage
    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.
  • by VisiX (765225) on Friday August 13, 2010 @09:49AM (#33238524)
    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.
  • by bo-eric (263735) 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?)

  • The problem (Score:4, Insightful)

    by AhabTheArab (798575) on Friday August 13, 2010 @10:06AM (#33238848) Homepage

    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.

  • by zero_out (1705074) 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.

  • by Ereth (194013) 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.

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