_+3=5+7=_: it depends what ‘equals’ equals

Diandra Pop quiz!  Answer the following before reading further:

A:  6+9=__+4

B:  __+8=12+5

C: __+3=5+7=__

D:  True or False:  6+8=3+11

E:  160=___

I've become painfully aware recently how sloppy communication can be.  I am surprised how many times I have to reply to an email asking for clarification because of unclear writing or simply not taking the time to think things through.  And that's with other scientists.

The GK-12 program I run at the University of Nebraska places graduate students in upper elementary, middle and high schools to work with teachers on improving math and science education for their students.  We spend an entire day of the week-long orientation discussing communication.  I break it down roughly into "scientific" communication and "normal person" communication.  That's probably not a fair breakdown, but we really have to emphasize to the students that, although it is perfectly OK to reply to a scientist's idea with "here's why that won't work", it's a death knell for the relationship if you do that with a teacher (or, often, a spouse).

At the end of the year, one of my students made an observation I wholeheartedly endorse.  "I like scientific communication better," she said, "It's just faster."  And clearer, I would add. 

Although Jennifer and I come from very different disciplinary backgrounds, I think one of the reasons we've hit it off is that we share the trait of wanting to use words properly.  Jennifer recognizes that scientists and mathematicians use words and symbols to convey very specific meanings.  If I use the word "velocity", she's likely to ask if there's a reason I didn't say "speed".  (Speed is a scalar, velocity is a vector.  Sometimes it makes a difference, sometimes not.)

Nowhere, perhaps, is the specificity of symbols more rigorous than in mathematics.  My mother was a graduate student in math, then economics, while I was in elementary and middle school.  I remember seeing her scribblings filling up scads of yellow legal pads and asking her once "when do I get to learn this language?"  And math is definitely its own language.  One of the biggest problems teaching (or communicating) science and math is that sometimes words mean different things in the discipline than they do everyday life.

But the equal sign should be an easy one, right?  It means, well, equal.

Apparently, American students have a much less clear idea of the meaning of the equal sign than their Chinese, Korean and Turkish compatriots.  A study by Capraro, et al in Psychological Reports (106(1), 49-53 (2010)), which draws on their previous data in Li, et al. (Cognition and Instruction, 26, 195-217 (2008) compares 6th grade students from different countries.  Both papers originate from the research group of Mary Margaret Capraro and Robert M. Capraro at Texas A&M University.  Incidentally, "et al." translates literally in Texan to "and them".

The results from the first two questions I posed from their study (6+9=__+4 and __+8=12+5) were surprising/appalling.  Only 28.6% of American students got these questions right.  The Chinese and Korean rates were in the 90+% range and the Turkish rates were 61% and 79% respectively.

As is often the case, we learn more by looking at the wrong answers than the right ones.  The first two problems were designated "Type A" and "Type B" – similar, except the missing numbers are on opposite sides of the equal sign.  The third problem – the one that lent itself to the title of this blog – is classified as "Type C" and provides a slightly different probe of the understanding (or misunderstanding).

For the "Type C" problem __+3=5+7=__, American students got the first blank right 23.8% of the time, while the rates for other students were 98.6% (Chinese), 86.5% (Korean) and 60.2% (Turkish).  Interestingly, the correct rates for the second blank were much more comparable:  86.7% (American), 97.9% (Chinese), 93.3% (Korean) and 86.0% (Turkish). 

What this strange disparity between the first and second blanks tells us, the authors argue, is that American students disproportionately don't understand the equals sign.  The most common wrong answer for the first blank was "2".  It is true that 2+3 = 5, but 2+3 definitely doesn't equal 5+7.  Almost 90% of the students recognized that "5+7=12", but a significant number of those got the first blank wrong.  This is apparently a common misconception among American students that isn't seen nearly as much in student from the other countries studied: the belief that the answer is the number immediately following the equal sign.

In their previous study, the authors used the True/False question "6+8=3+11?" to test understanding of the reflexive property of the equal sign.  Reflexive, which I had to look up, means a=a. The popular phrase "it is what it is" embodies the mathematical philosophy of reflexivity.  The educators doing the study,  though, realized that students with the misconception I mentioned above – the answer is the number immediately after the equals sign – would get this question wrong for that reason and not because they don't understand that a=a.

In their new study, they replaced that question with "160=___", and expected the blank to be filled in with "160".  But a number of students put an operation in that blank, like 80*2 or 40+120. Those answers are not wrong, but (strictly speaking), but they indicate that those students look at the equal sign as indicating that a mathematical operation is required and not solely as a representation of equality.

I've always thought of the equal sign as the pivot on a see saw.  Whatever is on the left has to balance with whatever is on the right. If I fill in the blank with a 6, there's only 9 on the left and 12 on the right, so the see saw isn't balanced.  OK, I can't draw a picture for the multiple equals signs on one linBalancedEquatione, but you get the idea.  Who ever thought that something as seemingly simple as 'equal' could be so complicated?

At this point, you might be thinking that this seems like a bunch of quibbling over a precise definition of interest only to the highly mathematical.  One of the original motivations for this study was the low performance of American students relative to their international counterparts on standardized tests like the TIMSS and the PISA tests.  There apparently isn't much explicit attention to the equal sign in middle school curricula and, the authors (along with other math educators) believe that not understanding the equal sign puts students at a distinct disadvantage when it comes time to learn algebra.  If you put 'x' in place of the blank, you realize you're actually doing algebra answering the questions I posed at the start of the blog.

These misconceptions carry over to physics.  For example, consider the force on an object falling:  F = mg.  The force gravity exerts on a ball falling through the air is equal to the product of its mass times the acceleration due to gravity (in the absence of air resistance – sorry, I felt compelled as a professor to add that.  I couldn't help myself.) 

Students identify gravity as a force, but a significant number of them also identify a force "F" in the above equation as distinct from the force of gravity.  The problem gets worse when there are multiple terms on the right-hand side of the equation due to multiple forces.

It is amazing that something so seemingly fundamental can so impact a student's education.  One of the (many) reasons I am looking forward to Jennifer's book is that I speak fluent calculus.  Imagine trying to explain to someone how to walk.  That's what me teaching calculus is like.  One of the best reasons for using peer teaching (students teaching each other) is that they explain things in ways I wouldn't have thought to use.  Listening to them explaining how they understand an idea helps me realize how I can explain it better.  It's research like this that reminds me that sometimes the better part of teaching is listening.

ADDED 8/13/10:  An interesting study notes the need for better prepared mathematics teachers, as well as a significantly strengthened math curriculum.  Jennifer and I have been talking a lot here recently about stereotypes. Although we've focused on those in the media, this article, by researchers at the University of Chicago in PNAS, suggests a scary chain:  Female first and second grade teachers who are anxious about math pass that anxiety along to their female students. More female students are likely to agree with the suggestion that boys are better than girls at math after being exposed to this anxiety, and the female students who did agree with this stereotype performed worse in math as the year went on.  Great article and PNAS makes the full text publicly available.

AND:  A Christopher E. Granade speaks on the topic of 'equal' – a very nice post focusing on the importance of relationships and how that is really at the base of math and science.

16 thoughts on “_+3=5+7=_: it depends what ‘equals’ equals”

  1. Thank you for this – I’ll be teaching physics this fall, and I will be using some variation of this mini-quiz to pinpoint these reasoning errors of the students.

  2. Thank you for this very interesting article. It drives me insane when my students write something like 3+5=8-7=1. I think we take for granted that they will understand the meaning of “=”, when according to the figures quoted above they certainly do not! I might use those five questions at the start of the article with my students this year.

  3. This is clearly a ‘hot topic’ at moment! I agree with you about precision in the use of language (actually the impact of teaching language on the learning of Mathematics is my main research interest!). I have already made comments about this on two other blogs: http://castingoutnines.wordpress.com/ Robert Talbert on “calculator syndrome” (as I call it) and http://georgewoodbury.wordpress.com/2010/08/11/thoughts-on-equal-signs/#comment-261 George Woodbury’s thoughts on simplification of algebraic expressions. Your readers may find them interesting too.

    Oh, and since you mentioned speed… does it annoy you when people talk about “driving at a high rate of speed”? Surely they mean acceleration, don’t they? Or, do they just mean driving very fast…

    Colin

  4. diandra@trivalent-productions.com

    Colin: Yes! I am an advocate for talking simple: “I was driving fast” is accurate and works just fine.

    I like: “I was turning the corner and I wasn’t even accelerating…” and anything Rusty Wallace says when he’s doing commentary during a race.

  5. I wouldn’t characterize this as a difficulty with understanding equals so much as a difficulty with operator precedence. If you placed parentheses, making, for example, (_+3)=(5+7), I’ll bet the failure rates change dramatically.

    As a long-time programmer and computer scientist, whenever I write code that is meant to evaluate such an equation, I put in the parentheses, it’s just simpler. I will furthermore add that the sort of thing that cox_dan points out above makes perfect sense if you are working on a calculator. It’s exactly the steps you might take in a calculation: you might press ‘3’ then ‘+’ then ‘5’ then ‘=’, showing ‘8’ on screen, then ‘-‘ then ‘7’ then ‘=’ again showing ‘1’.

  6. not all of us are observant enough to notice this, if you’re noticing what I think you’re noticing. the difference between boolean equals and the other kind. boolean equals is noted as “==” or an equals with three lines instead of two. this should be a simple and obvious topic because it has been studied for how many decades or hundreds of years? but it certainly isnt common curriculum. the most basic definitions of our most basic math operator.

  7. I was coaching a boy who had a bit of trouble with equations. So I got him to stand up, arms outstretched, and pretend to be a balance.

    Then I asked him to add, say +5, to one side, and then asked what’s happening? Then to ask him to add +5 to the other side, and asked again. This seemed to help.

  8. I guess that I had good math teachers in high school because they always taught that the “=” symbol read “true” not equals.

  9. Yes! This couldn’t be more important. Thank you. Having tutored HS math students, many read = as the enter key on a calculator–no understanding of the symbol having meaning.

  10. Interesting, but the conclusion about the 160= part really baffles me. Mind you: i’m just somebody who finished high school and who does nothing with maths. When I did the ‘test’ I also finished it with something like 160=120+40. Not because I expect some kind of operation after the blank, but because I thought ‘well of course 160=160! That is so bloody obvious. But this test obviously requires me to do some addition tricks, so, if the testers want me to do a trick, I’ll do it’

    I think it’s extremely logical to fill it out like that. When you do a test, you’re always second-guessing the people who put it together. It’s not because I thought the equal sign requires an operation, it’s because I thought the testers required an operation. It has nothing to do with understanding the equal sign.

  11. My answers:
    A. False
    B. False
    C. False
    D. True
    E. False

    Did I get them right?

    [Note: Given that the author of this paean to precision amusingly failed to actually ask a question, I substituted in my own:
    “Assume __ = 0. Are the following equations true or false?”]

  12. Maybe I’m just weird, but when I saw __+3=5+7=__, I initially assumed that the definition of + was what was really at stake, that some kind of cyclic group or modular arithmetic or something was involved. (Upon further reflection, that doesn’t work either, because it would be mod 3, but there are numbers larger than 3 in the equation.) It did not occur to me that the same value (represented in the problem’s symbology by an underscore) could stand in for different numbers on the same line of equation.

    Of course, it’s possible the problem’s instructions (which I didn’t see) hinted at that, with wording like “fill in each blank with whatever number will make the equation true”, or somesuch.

  13. 2+3=5+7=12 makes perfect sense to someone who is using a calculator. Since American kids use calculators for even the most trivial arithmetic, it’s no wonder they would see this as a reasonable math “sentence”. Kids in many places around the world still do arithmetic manually, and seldom see this kind of strung-together expression.

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