Misuse of parentheses for multiplication

Question

I'd like to raise the issue of constant misuse of parentheses in the U.S., and I'm wondering if anybody else shares the same feelings, has had the same issues, knows any history behind it, and can offer some thoughts on the subject. Let me explain what I'm talking about.

In elementary algebraic notation, i.e. when writing algebraic expressions, parentheses mean two things: order of operations and function notation; http://mathworld.wolfram.com/Parenthesis.html seems to agree with me. (There are all the other things, such as interval notation, matrices, etc, but I'm not discussing those here.) But here in the U.S. it is used all the time to enclose each factor in multiplication, so $5\cdot2$ would be written as $(5)(2)$. And while it's technically speaking not wrong, as it still means the same thing, it's redundant and pedagogically terribly harmful, imho. This way of writing multiplication is so persistent from early on and then everywhere, that students grow up believing that that's what parentheses mean: that their purpose is to enclose each individual quantity in multiplication. Not for grouping, not for function notation, but mostly just for that. Almost any of my students here (and I've tried asking a whole class, so it's kinda experimentally confirmed) will write the product of $2$ and $x+1$ as "$(2)x+1$".

In my teaching I always try to correct my students' bad writing habits. Here's a recent example. Solving an example on the board during my "College Trigonometry" class, I actually came to the need of setting up a product of something like (don't remember exactly) $2$ and $\sin x+1$, which I intentionally wrote as "$2\cdot\sin x+1$". I turned to my class and said: "By the way, this is wrong. Can you correct what I wrote?" One of the best students in the class immediately said: "It misses parentheses."

"Halleluja! My efforts haven't been futile," I thought to myself... until he continued: "$2$ must be in parentheses because it's multiplied."

It would have been just a little notational nuisance, had it not been a cause of so many real mistakes. The most obvious one is when people don't distribute multiplication, since there's nothing indicating the need to distribute. I mean, as a quick example, when instead of $2(x+1)$ someone wrote $(2)x+1$, and then they need to plug e.g. $x=3$, they'll get $7$ instead of $8$. Another one, that I believe stems from the same misuse of parentheses, is misreading function notation. Again, I've done this literally with thousands of my students, and most read "$f(x)$" as "$f$ times $x$" or "$\sin(x)$" as "sine times $x$".

I only have the experience of working in two countries. In Russia, my homecountry, nobody ever writes $(2)(5)$. (And mind you, its educational system has a truckload of problems of its own, but at least this is not one of them.) In the U.S., it's not just about the students, but it's all over the books, textbooks, and in almost everybody's math writing, including teachers and professors — which is probably why all students pick up this habit. I honestly don't know about the rest of the world.

So, back to the questions that I asked in the beginning. Can anybody explain where this writing tradition comes from? Does anyone agree that it causes some real problems in students' learning of math?

Update: Let me clarify my concerns a little bit. As others pointed out below, $(2)(5)$ is weird but not wrong. And I agree: it's not wrong mathematically, but it is wrong pedagogically. And although it still irritates me (to put it mildly), that's not the problem in itself. But $(2)x+1$ is actually wrong when it stands instead of $2(x+1)$. I apologize for not being clear enough, but here's what I meant: I believe that seeing $(2)(5)$ results in students' writing $(2)x+1$, and that's wrong, and that's the problem.

Answer 1

To answer the ultimate question ("Can anybody explain where this writing tradition comes from?"): It's explicitly taught that way by many U.S. instructors and textbooks.

Examples: From the otherwise excellent Martin-Gay Prealgebra & Introductory Algebra (sec 1.5):

The × is called a multiplication sign... The symbols ∙ and () can also be used to indicate multiplication.

Shaum's Outlines Elementary Algebra (sec 1-4):

The symbols for the fundamental operations are as follows... Multiplication: ×, (), ∙, no sign

A research presentation for the NCTM actually took not thinking that parentheses are multiplying as a sign of student misunderstanding (Welder, "Using Common Student Misconceptions in Algebra to Improve Algebra Preparation"):

Misconceptions: Bracket Usage -- Beginning algebra students tend to be unaware that brackets can be used to symbolize the grouping of two terms (in an additive situation) and as a multiplicative operator

I could go on (and probably more U.S. teachers teach this even more adamantly; I've never met anyone who opined otherwise). To answer the other question ("Does anyone agree that it causes some real problems in students' learning of math?"), then I would say: Yes, absolutely.

The most common mistake I see in this vein is that if one is taught both that "parentheses mean multiplying" and in the order of operations that "parentheses come first", then that combination results in many students thinking that juxtaposed multiplying is meant to happen first, e.g.: They will simplify $2(3)^2$ as $6^2 = 36$ and so forth.

Others will think that parentheses are entirely interchangeable with the dot notation and juxtaposition, e.g: when substituting $x = -2$ in the expression $3x^2$, they're prone to writing things like $3 \cdot -2^2$ or $3 - 2^2$, which they may or may not recover from on the next line. Even just writing $3 \cdot -2$ is nonstandard, of course, but if corrected they'll grouse about it as being easier than using parentheses.

As noted in a comment, I've written about this on my blog in the past on MadMath. I think it would be tremendous if we could flip this around and start teaching that it is juxtaposition that indicates multiplying, not the parentheses themselves (and therefore reserve more focus for parentheses grouping and giving precedence to any operation contained inside, including unary negation).

Answer 2

It is actually wrong to say that parenthesis means multiplication. In $(2)(5)$ it is the lack of an operator between the parenthesis that implies multiplication, NOT the parenthesis. The parenthesis are "needed" because $25$ means the number twenty-five, the parenthesis are purely for grouping.

"No operator means multiplication" is an extremely common convention that is much more useful to learn than "parenthesis means multiplication", which is just plain wrong, since it only means multiplication when there is either a multiplication operator (obviously) or no operator, which the first rule already covers.

So yes, I agree that teaching students something that is not true and unnecessary very likely causes problems for students learning math

Answer 3

I disagree that it is "terribly harmful".

Do not prevent them from writing $(2)(5)$. Instead prevent them from writing things that are actually wrong.

Thinking that $\sin x$ is $\sin$ times $x$ can happen (and does happen) even without parentheses.

I agree $(2)(5)$ looks strange, but if they can write $(4-2)(4+1)$, what is wrong with $(2)(5)$?

Answer 4

This problem starts in elementary school. In fact, I am one of the elementary school teachers that has started students along this path that you complain about. I have taught my 5th and 6th grade students that once they start using variables they should no longer use the multiplication sign since it might be mistaken for the variable x.

I teach my students that while the book might use a dot for multiplication the students should never write $5\cdot2$ since they might confuse it with $5.2$ From experience I can assure you that elementary students will not be able to distinguish between the multiplication dot that they write and the decimal point that they write. In a printed book it is much clearer.

As is standard in textbooks and the literature that I've read, I teach my students that they can use parentheses for multiplication symbols. I have tried to teach order of operations in great depth so that they don't think that $(2)5+1$ is anything other than $10+1$. In fact my students know that

$(5)2+1=10+1=11$ because they know that $5\cdot2+1=10+1=11$

I would suggest that a greater effort be made to teach order of operations in depth. Most of the elementary teachers that I know don't understand order of operations and many are still stuck on PEMDAS and are convinced that addition comes before subtraction since A comes before S. They believe:

$8-3+5=8-8=0$ and not $8-3+5=5+5=10$

I would also suggest that * be adopted as a multiplication symbol. It is easy for students to draw and widely used in spreadsheets. It is much easier for students to use than the multiplicative dot and would eliminate the problems of parentheses.

Answer 5

Another one, that I believe stems from the same misuse of parentheses, is misreading function notation

To me, this is the more serious problem that occurs. I don't necessarily have people saying "f(x) is f times x" but I do have people that then note $f(3)=6$ deduce that the function is multiplication by two. Once you start composition of functions (this even happened to me today), we get the question of $f(g(x))=f(x)g(x)$...

But as to a "solution", I have a different point of view on this. I agree that the ambiguity is terribly annoying and can be harmful; but natural languages have much more ambiguity, and we somehow get around that. Because we have a limited number of symbols and ways to indicate typical operations, and because much notation is ingrained due to many years of use, it is often better to teach what people will actually see in "real life" and provide good warnings.

As a non-parenthesis example, I will take $\frac{df}{dx}$. No one would suggest (I hope) that, historically speaking, this notation has not been very successful, and that it not only reminds us that derivatives come from difference quotients, but even (properly handled) allow very convenient manipulations for e.g. solving differential equations. Nonetheless, from the point of view of the question here, this is horrible notation! And some misconceptions arise from it. So we teach when you can use this or when not (at least if we are mathematicians and not physicists).

Another example is exponential notation, where I often have students who are quite confused why in $e^{x^2}$ the "inside function" is $x^2$, and exactly how things are related. Better, from this point of view, is $\exp(x^2)$. But there are other natural advantages to exponential notation, so we use it, and - hopefully - point out the ambiguity and give good exercises or activities to highlight the "right" answers.

So to bring this back to parentheses, I think that the "right" solution is to acknowledge this ambiguity and try to resolve it. One way in which I often do this is to explicitly write $\cdot$ or $\times$ as appropriate; another way is to have activities which highlight this difference and then really engage each student individually on it. This is pretty labor-intensive, and won't work for every situation. But simply avoiding the parentheses altogether - for now - is worse, because students simply will not interpret $3+2x \cdot x+1$ as anything other than $4+2x^2$, as as you point out grouping is what they are for. In fact, I usually have the opposite problem of getting them to use parentheses to indicate this is $(3+2x)\cdot (x+1)$. So that is what I recommend, in this context. Seeing $(2)(5)$ on occasion is a small price to pay for them understanding the grouping bit; there isn't really any other interpretation possible, at least.

All that said, if you have good ideas of how to resolve this systematically from your previous experience, so that no one writes $(2)(5)$ but still knows what to do with $x^2(x+1)$, it would probably be a good thing. (And hopefully that doesn't include replacing $2\cdot 5$ with $2\, . 5$ as the British do!) I'm just not sure how to do that, and I have to teach the students I actually have.

Answer 6

There is a perfectly good reason to write $(5)(2)$ rather than $5\cdot2$, which is that the dot can easily be confused with a decimal point. I believe Russians use a comma as a decimal point, so it makes sense that Russians would not see any danger of error in writing $5\cdot2$.

Also, scientists and engineers typically think of numbers as having units (unlike most mathematicians, who seem to think of units as part of the definition of the number), and therefore consider it scientifically illiterate to write numbers without units. So suppose we're going to multiply 5 m/s by 2 s. Writing it as $5\ \text{m}/\text{s}\cdot2\ \text{s}$ is ugly and hard to read, so we write it as $(5\ \text{m}/\text{s})(2\ \text{s})$. I find, e.g., $2\pi r=2\pi(1.2\ \text{cm})$, much more readable than $2\pi r=2\pi\cdot1.2\ \text{cm}$. Even for unitless numbers, an expression like $2\cdot7.8\cdot0.134$ looks like a hot mess to me compared with $2(7.8)(0.134)$

The disease that I've been seeing a lot within the last 5 years or so is the use of parentheses in totally unnecessary ways like this:

$(2)(x)$

$\sin(x)$

$((2)(x))/((y)(z))$

I suspect this is actually because of the increasing use of software on computers, phones, and calculators that has fancier symbolic math capabilities than most calculators used to have. Math and science students are typically entering their homework answers into computer software systems such as Mastering Physics or MyMathLab.

Many such systems require parentheses for function evaluation.

Computers don't have common sense or understanding of context or meaning, so they can't interpret expressions like $\sin\omega t$ the way any engineer would (as $\sin(\omega t)$, not $(\sin\omega)t$).

Also, students will have had the experience of having the software interpret their input according to the normal rules of precedence of operations, just as a human would, but the result isn't what the student expected because the student doesn't understand the order of operations. For example, the student intends $a/(b+c)$ but types $a/b+c$. Instead of learning from this experience that they need to get more fluent with the order of operations, they just decide that it's safer to use lots and lots of parentheses.

Another thing that's going on now with systems like Mastering Physics or MyMathLab is that many lazy teachers take them as excuses to avoid ever reading students' work and writing comments on it with a red pen. Therefore when students make stylistic mistakes such as $(2)(x)$ or $\sin(x)$, they never get any feedback from a human telling them that their style is wrong.