Issues with "equals", where does this come from and how do I combat it?

63

20

An issue I see with students a lot is abuse of the equals sign. For example, one problem asked "what is the degree of the polynomial: $\text{polynomial}$?", and I got answers like "$\text{polynomial}=3$". I tried to explain that no, that polynomial is NOT equal to 3, the DEGREE is equal to 3. They had a difficult time understanding what was wrong.

Another example, same student. The problem was something like "evaluate $5\times 2+5\times 7$". The student keenly asked "since there's a $5$ in both, can we divide by $5$?". The question of course is, "divide WHAT by $5$?". I could have said "we can let $x=\text{blah}$ etc." but that would be too confusing. I just said "since we don't have an equation, we can't divide, but what we can do is factor out a $5$ to get $5\times(2+7)$". They were satisfied with this.

I know it sounds pedantic, but conceptually equality is perhaps the most important concept in algebra to understand. Not to mention this can lead to real mistakes. For example I saw on a question like "expand $x(x+1)(x-1)$" the following answer:

$$x^2+x-x-1$$ $$=x^2-1$$ $$=x(x^2-1)$$ $$=x^3-x$$

They happened to get the right answer, but in doing things like that it's really easy to make mistakes. I tried to explain that lines 2 and 3 were clearly not equal, and again it just seemed so lost on them.

Why is this so difficult for students, and how do I combat it? What are they hung up on?

Zachary Selk

Posted 2015-04-27T03:12:02.390

Reputation: 970

6

Related: See my answer at http://matheducators.stackexchange.com/questions/1040/how-to-cure-students-from-the-idea-that-root-and-squaring-are-identity-operators/1058#1058, and the comments on that answer, for a further discussion of the "operator" conception of equality.

mweiss 2015-04-27T16:54:49.280

1Regarding the last example - simply exchange x with some value (say, 5) and show them, that 24 does not equal to 120 :)Spook 2015-04-28T10:23:40.410

2My high-school math teacher used to explain equality with a pair of scales. Things are only equals if both sides "weigh" the same. You can only use an equality sign if the scales would be balanced. When classmates made the mistake your student makes, he would force them to draw a pair of scales and put the values on there, then asked them if their drawing made any sense. It never did :)Kevin 2015-04-28T11:18:15.720

3To complicate things, in computer science it's perfectly okay to say $x=x+1$. There "equals" means "assign the value of".Aeryk 2015-04-29T13:51:05.850

1I have tried to combat this with my elementary students who when rounding 28 to the nearest 10, will write 29=30. I point out that they are not equal and suggest using an arrow 29 => 30 to show that they are going to the next step.Amy B 2015-07-16T00:38:49.303

If I am doing a problem in that way (which would only be in one-on-one work, never with a whole class) I would either omit the sign entirely, or use an arrow instead.Wildcard 2015-10-27T05:59:49.637

3On the "how do I combat it" question: you have to grade on it. Every time an equals sign gets misused on a test, points come off. Every time a student writes a falsehood (that things are equal when they're not), points come off. State this expectation clearly up front. I think that proper writing of the language is the most valuable thing we can share. If they don't get feedback with points on the line, they'll never attend to it (and some still won't).Daniel R. Collins 2015-11-29T19:49:16.450

Also, I would advise please not to use arrows instead, for two reasons: 1. the student that try some math-heavy higher ed then get totally confused when we try to use "=>" precisely to denote logical implication; 2. it makes more sense to use a sentence explaining what is done or what happens. Student should not think maths is written in a sequence of symbols, it is written in a language. Sure, having them write sentence that make sense is hugely difficult, but for one main reason: it does not hide the depth of misunderstanding that happens also when it is covered by sloppy notation.Benoît Kloeckner 2017-04-03T14:54:57.323

Answers

66

A surprisingly large number of students don't know what the equals sign means. Their understanding of the symbol "=" is essentially operational, not relational — they think it means "the next step" or "the answer" or is an instruction to perform some operation. Knuth et al. ("The importance of equals sign understanding in the middle grades", Mathematics Teaching in the Middle School, vol. 13, no. 9, May 2008) studied middle school students' understanding of the equals sign and identified this misunderstanding as both widespread and strongly correlated with inability to correctly solve basic algebra problems. This serious conceptual error often doesn't correct itself without being directly addressed; I've seen university calculus students make the same sort of mistakes.

Where does this error come from? As Knuth et al. remark:

Researchers have argued that the operational view of the equal sign is largely a by-product of students’ experiences with the symbol in elementary school mathematics (e.g., Baroody and Ginsburg 1983; Carpenter et al. 2003; McNeil and Alibali 2005). During elementary school, students typically encounter the equal sign in number sentences that have operations on the left side of the equal sign and an answer blank on the right side (e.g., 5 + 2 = __, 11 – 4 + 1 = __). In solving such “operations equal answer” equations correctly, it is not really necessary for students to think about the equal sign as a symbol of equivalence; rather, students need only perform the calculations on the left side of the equal sign to get an answer. As a result, students associate the equal sign with the arithmetic operations performed to get a final answer.

There are even examples of exercises in elementary school mathematics textbooks that make the same error, using the equals sign in a way that can only be interpreted operationally. (I can't recall where I saw this; I'll add a link to an example if I find it. Edit: mweiss found this example, linked in the comments; that's not the same one I remember seeing, so there might be several textbooks doing this, or it could be from a different place in the same book.) Such textbooks have probably done quite a bit of damage.

Since conceptual understanding of equality is so essential to all mathematics beyond elementary arithmetic, I think it's advisable to have students solve problems that directly address the meaning of equality. I have no experience with teaching at that age, so I'll leave coming up with appropriate exercises and lessons to those who do. Instead, I'll just explicitly state the concept of equality that students must deeply internalize: "S = T" is the statement that S and T are literally the same thing, and consequently are indistinguishable in every way and interchangeable in every context. Syntactically, this amounts to the fact that any statement involving S is true if and only if the statement formed by replacing every instance of S with T is true.

As a side note, I suspect this is also related to an overly procedural concept of what other mathematical symbols mean; I'd guess many students who don't know what the equals sign means also interpret "1 + 2" as an instruction ("add these numbers"), not as a number. Thus, "1 + 2 = 3" is being interpreted as "if you perform the addition procedure on 1 + 2, you get 3", not as "the number 1 + 2 is the same thing as the number 3".

Daniel Hast

Posted 2015-04-27T03:12:02.390

Reputation: 4 154

10

A related (less reader-friendly) Knuth et al paper is Does Understanding the Equal Sign Matter? Evidence from Solving Equations.

Benjamin Dickman 2015-04-27T04:55:56.387

2Wow this is an amazing post, thank you. I meet with some students tomorrow and I might want to spend some time talk about equal signs in this light. Specifically addressing the issue, maybe bring up the example of "what does this sign mean" in 3+4=7. I read both the reader friendly and non reader friendly paper, both were extremely enlightening. Thank both of you! Edit, I just realize that the students that have issues with equality have issues with inequality as well! I'm sure there's a connection.Zachary Selk 2015-04-27T06:38:02.297

1So early curriculum often treats equality as operational = students learn to think of equality as operational = students write nonsensical answers in college because they lack a conceptual understanding of equality? ;)Michael Joyce 2015-04-27T15:04:48.327

7

Regarding examples of the "operational" error in elementary curricula: I posted an example of that in the comments at http://matheducators.stackexchange.com/questions/1040/how-to-cure-students-from-the-idea-that-root-and-squaring-are-identity-operators/1058#1058. The example was at https://drive.google.com/file/d/0B4wpC387vwzeeE04YXhrNkhRcG8/.

mweiss 2015-04-27T16:56:27.973

18Another good point in the comments at that other question is that strings like 3 + 5 = 8 + 2 = 10 + 5 = 15 are perfectly accurate if read as a transcript of a calculator session. The 8, 10, and 15 are what the calculator tells you. Everything else is what keys you press to get those answers. And it's all in chronological order. I'm surprised calculators weren't mentioned in the research papers above, since the "operational" interpretation of = is exactly the meaning of the calculator's = key: "Perform operation and display answer now".Wumpus Q. Wumbley 2015-04-27T17:07:55.113

@mweiss wow that's actually horrible! In this light, that book is actually really detrimental. Thanks for sharing.Zachary Selk 2015-04-27T19:00:08.073

11I've encountered this problem often enough in university classes (and not just freshman calculus) that I often try to prevent it by loudly saying something like "Don't write $=$ unless you really mean that what's on the left is the same thing as what's on the right." Such prevention seems to do some good, but sometimes I need to repeat it several times during the semester.Andreas Blass 2015-04-28T01:27:20.997

8Even if x=y is explained well enough, will the typical student ever encounter an explicit definition of x=y=z as meaning x=y AND y=z? Or do we just start using it and expect everybody to get it? If it's not officially defined then abbreviating a+b=c AND c+d=e as a+b=c+d=e isn't wrong, it's just an alternate notation. Defining it precisely is only slightly tricky (roughly: the first term of the expression between the equal signs is the same as the thing on the left of the first one). If a single equal sign is worthy of a formal definition, then so is a pair of them chained together.Wumpus Q. Wumbley 2015-04-28T12:50:09.920

3Isn't this essentially just a matter of operator precedence? (a+b)=(c+d)=(e) vs (((a+b=c)+d)=e)Random832 2015-04-28T18:54:37.477

1@Random832: I think failing to recognize that equations are statements is part of the problem. To students who know what the equals sign means, there's only one way to interpret "a + b = c + d = e" — adding a statement and a number doesn't make sense, so "(a + b = c) + d" is clearly an incorrect parsing. (A more subtle issue is that many programming languages use "=" for assignment, not for equality, and assignments return the assigned value rather than a truth value. However, I doubt this is the issue here; probably very few of the students making these errors know a programming language.)Daniel Hast 2015-04-28T19:26:00.383

2@DanielHast It certainly seems reasonable to throw an equation in parentheses and consider the resulting clause equal to all of its sides, even without any analogy to assignment in programming languages (and indeed no real programming languages work that way, but it's a perfect fit for the calculator logic). I considered putting x((x+1)(x-1)=x^2+x-x-1=x^2-1)=x^3-x in my answer below as a suggested notation. None of the sides of the inner equation are a simple variable that could be considered to have been "assigned" - it is a formula in various steps of being algebraically manipulated.Random832 2015-04-28T19:30:03.680

@Random832: Fair enough — I suppose the usage you've described is reasonably consistent. However, in light of the Knuth et al. study, I'm inclined to think that there's often a bigger underlying problem than just not knowing the standard notation, given the correlation with real difficulty in getting the right answer at all (not just writing it in an accepted format).Daniel Hast 2015-04-28T20:48:27.737

4I've also heard -- and maybe it's in one of the sources mentioned here, but I didn't see it explicitly on first glance -- that (in line with an operational understanding of the equals sign) many children understand perfectly that 1 + 2 = 3, say, but would think that 3 = 2 + 1 makes no sense at all.Carl Offner 2015-05-03T02:16:12.653

2I agree with Wumpus Q. Wumbley and Random832. Actually, 2+2=1+3=4 with equality interpreted as relation is not true even with correct operator precedence. That's ((2+2)=(1+3))=4 that is (4=4)=4 that is true=4 and it's false even if we assume that true=1. We have to assume that = can mean both is and that is. Two plus two that is one plus three is four. Or: "Take 2+2=4 apples" reads "Take two plus two that is four apples", but "We know that 2+2=4." reads "We know that two plus two is four.".BartekChom 2015-05-08T11:38:30.980

@AndreasBlass: I have a hard time not reading your statement as: "Don't write" equals "unless you really mean that what's on the left is the same thing as what's on the right." -- Which probably means I'm OK.Torsten Schoeneberg 2017-11-04T19:20:55.743

13

I can't add much more than what has been stated already. But I come from a programming background, so maybe this example may add a different perspective (or explain some problems interpreting equals for other students).

In many programming languages, statements like this are perfectly acceptable: x = x + 1

In this case, the programming language DOES treat the equals sign as an operator. (You calculate the right-hand side of the "equation" then assign the result to the left-hand side.) If your student is a programming hobbyist, that may help explain why they are confusing the meaning of the programming context vs. mathematical context.

In Daniel's example ("e.g., 5 + 2 = __, 11 – 4 + 1 = __") this could sort of be interpreted as an operator: The underscore is being SET to the value on the left-hand side.

Some programming languages make this distinction by using == for the "relational" meaning, and = as the SET operator. (Other languages require the SET or LET keyword when = is used as an operator: SET x = x + 1.

Early hobbyists may learn when to use one symbol vs. the other in a habitual way without fully understanding the conceptual difference.

So: perhaps it may be better to emphasize that the equal sign actually DOES have two meanings (at least in some contexts), and to emphasize that the mathematics you're teaching only uses the relational aspect.

Randall Stewart

Posted 2015-04-27T03:12:02.390

Reputation: 521

2You may want to note that yet other languages, such as Pascal, use "=" for the relational meaning, and ":=" as the SET operator. I prefer this, since it lets the standard equals sign have the standard mathematical meaning and lets the non-standard assign operator have a non-standard notation.Rory Daulton 2015-08-06T20:55:57.577

Rory-I appreciate the feedback. Reminiscing about programming caused me to think that maybe the student understands the semantics of the equals sign perfectly well. Maybe she needs guidance on explicit syntax. She probably understands how to write the equals sign when the left hand side and the right hand side are all on the same line. Maybe she just needs to be told that IF the line STARTS with the equals sign (there's nothing on the left!), then the expression on that line must be exactly equal to the expression in the line above it.Randall Stewart 2015-08-06T23:51:33.777

2Programmers know much better than non-programmers what the equals-sign means, precisely because they understand what is an operation and what is a truth value (otherwise they cannot use if). Moreover, it is extremely worrying that many math students do not want to know any programming at all, and because of that they have simply zero appreciation for logical reasoning nor notational precision. You are right that the equality chain has a very specific meaning, which again I've never seen any teacher in my life explicitly teach this! (Except myself, so far.) Why not put it in your answer?user21820 2016-03-23T16:10:29.430

@RoryDaulton: As an ex-Pascal programmer I was half-expecting somebody to come forward with that comment (hopefully I can restart with Delphi one of these years). I once discussed this with a C-programmer. Their supporting argument for doing it the C-way ("=" is an assignment, "==" is a comparison) was one of economy. The assignments outnumber comparisons in a typical program by a fair margin. Therefore it makes sense to make assignment simpler to denote.Jyrki Lahtonen 2016-09-26T06:35:55.793

1When a math teacher says "Let $P(x) = x^{2} + 1$", the equals sign is being used operationally, as the assignment operator, essentially in the same way as in a programming language like C. The same statement makes no sense with the arguments reversed. So mathematicians use the equals sign in the operational sense too.Dan Fox 2017-04-04T12:45:52.210

@DanFox: The "Let" is important there. To continue the programming reflection, in the original BASIC language, that was required syntax for assignment.Daniel R. Collins 2017-08-11T06:47:09.680

9

I believe you are mixing up three problems here. One is the mistaken idea that "mathematics is all compact, weird symbols, if I don't use them to abbreviate my text, it's wrong" (your $polynomial = 3$ example), there is the "next step" interpretation, and finally the "assign a value" from programming.

vonbrand

Posted 2015-04-27T03:12:02.390

Reputation: 8 980

4I'm glad you brought up the first point; it certainly seems like many students don't realize that explaining one's reasoning using words (and complete sentences) is not only acceptable, but often preferable.Daniel Hast 2015-08-12T05:00:33.900

1

I asked THIS QUESTION in MSE and was soundly throunced for asking such a non-mathematical question. I am glad someone referred this question to me.

Steven Gregory 2016-03-16T08:32:00.557

@StevenGregory, if you are referring to the comment on your MSE meta question with the sentence "That's not a mathematical question," I interpreted the comment as saying that the MSE question you were talking about was not a mathematical question.Joel Reyes Noche 2016-09-23T01:05:52.750

@JoelReyesNoche - You are correct. At the time, I saw that problem and was upset at the abuse of the equals sign. I picked the wrong place to vent.Steven Gregory 2016-09-23T15:38:06.380

8

I think in your expansion example this is caused by wanting to solve one subpart of the problem first. In essence, they're trying to expand $(x+1)(x-1)$ first as a "subroutine" and then substitute it back into the original formula, but lacking a clear way to delineate this from the solution to the main problem. They chose to mark the substitution with an equals sign - what it is equal to, of course, is the original formula. Your problem with it is, reasonably, that it is not equal to the immediately preceding step. Line 3 isn't equal to line 2, but line 2 is equal to line 1, and line 3 is equal to the original problem statement. They don't understand that with this kind of notation, the implied left-hand-side of the equals sign is always the immediately preceding line.

They didn't happen to get the right answer, and it's condescending for you to suggest this. They knew exactly what they were doing in the substitution step, they just didn't know how to write it. You need to teach them the correct way to write this.

I might write this as:

  x (   (x + 1)(x - 1) )
    ( = x² + x - x - 1 )
    ( =     x² - 1     )
= x³ - x 

I don't know how to typeset this, the outer sets of parentheses are intended to be a single large pair. And, frankly, if I were merely trying to solve it for myself, I might not write them either - the fact that the equal sign on the final line is intended to the left relative to the others would make it clear enough for my own understanding.

Maybe their answer was likewise arranged on the page in a way that they thought made it clear what the "up-hand side" of the offending $=$ sign was meant to be. Or maybe they didn't think it mattered what their intermediate steps looked like, since the question was "expand ..." rather than "prove the expansion of ..." - it was never clear to me as a student when "show your work" meant being graded on procedure and when it simply meant "prove you didn't cheat".

Random832

Posted 2015-04-27T03:12:02.390

Reputation: 319

1I know exactly what they were doing. Clearly they wanted to do a subroutine. I wasn't being condescending. What I was saying is that in this particular case, it's VERY easy to make a mistake and they HAPPENED to not make a mistake.Zachary Selk 2015-04-28T21:08:56.823

1In regards to doing "subroutines" when doing or showing calculations by hand, I normally select which subexpression(s) to do subroutines on with an under- or overbrace or bracket. After simplifying each subexpression in its entirety, I'll usually "tidy things up" by replacing each subroutine'd expression by its simplification and continuing from there. Example:

$$x\underbrace{(x+1)(x-1)}_{\begin{subarray}{l} =x^2+x-x-1 \ = x^2-1\end{subarray}} = x(x^2-1) = x^3-x$$ – benguin 2017-04-10T13:31:14.537

1However this technique/notation is a bit clunky if the subexpression is difficult to "access" (say due to lots of nested fractions, radicals, exponents, etc). Assuming the overall expression is nice though, this method can even be extended with subsubexpressions, etc.benguin 2017-04-10T13:34:01.437