# Are there any negative consequences in applying operations/functions to a whole equality?

Some of my students solve equations not by applying the same operations on the left and right sides of an equation, but by applying the operation to the whole equality. For example, they may write something like this:

$$\sqrt{x-1} = 3$$

$$\left(\sqrt{x-1} = 3 \right)^2$$

$$x-1 = 9$$

$$x=10$$

instead of the usual second line $$(\sqrt{x-1})^2=3^2$$. I have never seen such notation (I have been educated in France mostly). So my first reaction was to consider it incorrect and ask the student to write it the "right" way.

On second thought, it seems that the notation is sound if we define $$f( LHS = RHS )$$ by $$f(LHS) = f(RHS)$$.

Still, I have a bad impression any time I see a student using this notation.

Should I prohibit my students to use this kind of notations?

Or, to make it less opinion-based, are there any objective (in the mathematical, logical, and/or pedagogical sense) reason I should ban the notation in my class?

• There is a similar question asked in How to Write Steps of Solving Equations? at which e.g. I posted my preferred approach to solving an equation like the one you have here, link. – Benjamin Dickman Dec 31 '18 at 2:38
• I believe it is better practice to write it how you prefer, not as the kid does. Just think the nonstandard notation will lead him into errors. Also,hard to communicate (since it is nonstandard). – guest Dec 31 '18 at 3:54
• +Here is, if not a potential drawback, then at least a wonder: In the example provided, the operation of squaring both sides is explicitly notated; indeed, H Towsner points to this and says, "it identifies what the actual step is." But, what about getting from line 3 to line 4? The imagined student has added $1$ to both sides, but this step is not explicitly identified. And so I am curious: Which steps are to be indicated? Squaring, but not addition? What about taking a square root? What about dividing both sides by $|x|+1$? (Etc.) Just some food for thought! – Benjamin Dickman Dec 31 '18 at 5:31
• To me, the biggest problem is that this signals the student not understanding the essential meaning of the equal-sign: it evaluates to a truth-value, not a numerical quantity. Putting it inside parens implies evaluating that truth value and then squaring it (which, taking a generous interpretation as a Boolean algebra statement, is just an identity operation). If they transition to computing and think that $(a = b)+1$ is the same as $a+1 = b+1$ then they're in for a rough time. – Daniel R. Collins Dec 31 '18 at 6:18
• I would say: do not write $\left(\sqrt{x-1} = 3 \right)^2$. Instead write in words: "square both sides". – Gerald Edgar Dec 31 '18 at 14:35

The notation your students are using isn't wrong. It's perfectly clear - anyone who knows algebra instantly knows what they mean - it's not particularly confusing, and it isn't going to quickly lead to a dead end where it's incompatible with upcoming material. Indeed, there's at least one good argument for it over the conventional notation: it identifies what the actual step is, rather than requiring the reader to look at both sides of the equation and compare them to realize what's happening.

It's just not the conventional notation. There might be a good reason for that (I haven't thought through the implications of using your students' notation very far), but if there's a reason not to use it, I don't think it's quickly accessible to your students.

So you can't really explain to your students why they shouldn't use that notation other than that it's not how other people write. That creates some problems:

1. student often have trouble reliably telling the difference between conceptual mistakes and notational mistakes, so focusing too much on notation can be very distracting,
2. relatedly, having to write things a certain way imposes a real cognitive cost on students, when they're already trying to learn the material, and
3. students get very frustrated if they feel like they're not getting credit for understanding material just because it's not expressed in the right notation; this can feed into a sense that math is just a list of arbitrary rules to be memorized, which discourages them from trying to understand it.

It might still be worth it to enforce the use of conventional notation. Your students certainly need to be able to read conventional notation, and at some point, if not now, they'll need to be able to write it.

(It's worth comparing to the similar tradeoffs when children misspell words in their writing - on the one hand, children eventually need to learn to spell correctly, while on the other hand, when children are first learning to write, obsessing about spelling everything correctly bogs children down so much that they just learn to hate writing.)

But you should consider whether it's worth the cost, and how you can enforce prohibiting it in a way that doesn't conflate notation errors with math errors or discourage students. (For instance, if your situation makes it feasible, you might allow rewrites of notation errors.)

• One possible, arguably “more correct,” meaning of $(a = b)^2$ is squaring the result of $a = b$, in this case, $\text{true}^2$ or $\text{false}^2$. That doesn’t make much sense, but if the operation were something other than squaring, or if we were in a system where $=$ isn’t strictly a logical operation so we weren’t talking about $\text{true}$ or $\text{false}$, it might, and then from there consistency might argue against this notation. Also, this makes exponentiation “distributive” over equality, which might have some implications. $(a+b)^2\implies a^2+b^2$ certainly is a common mistake... – KRyan Dec 31 '18 at 18:28

This seems to me an interesting question, with variants in many contexts, whose answer is not self-evident.

The notation $$(A = B)^2$$ should be considered problematic because it is undefined (another way of saying "nonstandard") in the same sense as undefined behavior in a programming language is undefined - it might compile, it might be interpreted properly, but there is no guarantee that it will be understood. Mathematical syntax functions as a shorthand for indicating to who reads it what operations were performed or should be performed to pass from one statement to the next. It works best when the translation is straightforward (when correctly done).

A square amounts to multiplying something by itself and generally for this to make sense one has to operate within some mathematical framework (algebraic structure) in which multiplications are defined - one needs some sort of binary operation on the set of things that can be multiplied. What is the square of an equality? Usually one considers an equality as having a truth value - $$0$$ or something nonzero - and in fact it makes sense to square a truth value - in a programming language such as C code of the form $$(A == B)*(A == B)$$ is syntactically correct and would have a value equal to $$0$$ if $$A$$ and $$B$$ are unequal and something nonzero (usually $$1$$, but in principle dependent on the compiler) in the case $$A$$ and $$B$$ are equal. However, note that, although coherent, this way of interpreting $$(A = B)^2$$ is not what the student has in mind in this case! The problem is that the correspondence between the notation $$\,^2$$ and the operations to be performed is not the usual one indicated by an exponent.

The student has in mind to indicate notationally the following reasoning: Since $$A$$ equals $$B$$, $$A^{2}$$ equals $$B^{2}$$. It captures far better the sense of the reasoning to write $$A = B \implies A^{2}= B^{2}$$ than to write $$(A = B)^{2}$$. Often students are taught to write simply $$A = B\\ A^{2}= B^{2}$$ with no symbol indicating logical implication, such implication being indicated by passing from one line to the next. (Since the implication is left implicit many students do not realize it is there. Such a student thinks of a pattern of operations, rather than a series of implications.)

Summary: what is problematic about writing $$(A = B)^2$$ is that the exponent does not correspond directly to squaring something in the sense of multiplying something by itself. Rather it is something that has to be unpacked and interpreted (like the notation for a line integral) if it is to be evaluated - and, moreover, there already exists a more easily and more directly interpretable notation for indicating the same operations.

One encounters things like this in many contexts. In linear algebra one sees often identification of $$1 \times n$$ row vectors, $$n \times 1$$ column vectors, and elements of $$\mathbb{R}^{n}$$. Many a successful engineer/physicist makes such identifications regularly, yet making such identifications usually indicates that someone does not distinguish well between equality and isomorphism. When teaching a class where it matters to understand this distinction, I generally start out by advising students about the incorrectness of identifying two things that are not equal, marking it but not taking off points, but later in the semester, when the error is repeated, I take points off (it is never a substantial deduction, usually only a marginal one). The experience is that students who don't eventually manage to understand the distinction between equating and identifying rows and columns have trouble with many other aspects of the course, so that insisting on such a trivial point is not as formal as it may appear to some.

Another similar example from linear algebra: one calculates ranks and determinants by applying elementary row (and, for determinants, column) operations. In the case of rank computations the matrices produced are (tautologically) equivalent via row operations but not equal, while in the case of determinants there results a string of equalities. Many students write $$=$$ in both cases, and this usually signals conceptual deficits that are fairly serious in the context of learning linear algebra. Such students are generally memorizing sequences of operations and have very little understanding of what those operations are doing (e.g. that row operations correspond to operations on equations), or why they are being performed (e.g. eliminating redundancies/dependencies). One understands what such a student is doing, such a student might very well obtain the correct answer to the particular formal exercise being solved, but such a student almost always does not understand what he/she is doing.

Writing $$(A = B)^2$$ in place of $$A^2 = B^2$$ seems to me somewhere in between these two examples and how to react to it depends on the context in which it appears. Objecting to it is less fussy than objecting to identifying row and column vectors, but the possible conceptual problem underlying it is not so severe as in the case of writing $$=$$ between matrices related by row operations. My approach would be to mark it in red and explain in detail what should be written, but to take off no points (since in this case it seems the student did compute correctly, just used nonstandard notation to communicate the computations). If the nonstandard notation is used repeatedly, I would probably progress at some point to taking off points. I say "probably" because exactly how to react surely depends on the context (the student, the class, the goals of the class, etc.). (If a university student writes this it is an indicator of low probability of success in university level math classes, but when written by a 10 year old it could even be an indicator of an autonomous and active mind - inventing reasonable notation is not so trivial).

• When I first saw the OP, I thought about extending every known operator (e.g. ², $*$, $+$) homomorphically to equations. Indeed $(A=B)+(A'=B')$ should be interpreted as $A+A' = B+B'$ and $(A=B) * (A' = B')$ as $A * A' = B * B'$, which is the same as applying ² in case of equal equations. While this alleviates the problem you mentioned, this view stemming from homomorphisms is probably not shared by a first-year student. – ComFreek Jan 1 '19 at 8:11

My experience is that students resent having to write out all the steps in algebra when they can do so much of it in their head. Therefore shorthand is a kind of compromise that allows them to show their work without writing everything out. These students like to text and abbreviate as much as possible. I would allow it while emphasizing that it is personal shorthand that wouldn't be used when writing mathematics correctly.

The problem is that you want them to learn the correct notation. Perhaps you could allow it but regularly give exercises where you give examples of shorthand that the students have used and have the students write the notation correctly. Shorthand quizzes (translating shorthand to notation) can send the message that this is important even if you don't insist on it all the time.