# How to teach brackets?

I was taught in a school that one has to use different brackets in expressions like $\{[(3+4)\cdot 4]^4\}^{1/2}$ to denote the order which subexpression is evaluated first. But can this be recommended in current mathematics? I guess no as one can create arbitrary complex expressions such that one would require arbitrary many different bracket notations. Also, I think that is good to teach that $()$ is for evaluating order, $[]$ is for floor function and matrices and $\{\}$ is for the set notation. So is it wrong if I suggest students to use the notation $(((3+4)\cdot 4)^4)^{1/2}$

• One reason that people use parentheses (generally like this, though [on occasion] with further symbols) is to help the reader match each opening parenthesis, (, with its corresponding closing parenthesis, ). But, in fact, the proper use of parentheses should not allow for ambiguity; so the final notation that you suggest is the norm. (Nevertheless, there is an option to use larger parentheses in TeX; precede the symbol with \big and compare, e.g., $\big( \text{ and } ( \ldots ) \text{ and } \big)$ ...) Oct 3, 2015 at 20:27
• I personally agree and would only use $()$ as the different brackets distract me. There is a lot to say for 'form of argument' . Id always go for a square root over a power half whenever possible for example.
– Karl
Oct 3, 2015 at 21:32
• @BenjaminDickman, LaTeX prefered usage is \left( closed by \right). They can be used with all sorts of "parentesis", even non-matching ones, and the sizes get automatically adjusted to what is "inside". Oct 3, 2015 at 23:27
• For purposes of priority, it's inner to outer. No nead for different shape brackets. Oct 4, 2015 at 1:18
• @beginnertutor: The way you were apparently taught is very nonstandard. Among other the issues others point out, consider that when shifting to computer programming one will have nothing but regular parentheses, so it's best to practice with the nested parentheses as soon as possible. Oct 4, 2015 at 21:13

It is definitely not wrong to teach your students to just use (). I would possibly argue the converse. Your experience at school flys in the face of convention. Teaching your students that way will not prepare them well for meeting the rest of the world.

• Your experience at school flys in the face of convention. Teaching your students that way will not prepare them well for meeting the rest of the world. This is incorrect. See my and Thierry's answers below.
– user18187
Jan 16 at 7:49
• @user24096 I believe Jessica B is referring to the "convention" that the OP mentions of always having to use different types of brackets.That's way too strict and indeed would be strange to teach. Others have taken their disagreement too far and argued that we should all use only parentheses to help the students, which is where we both disagree. Jan 16 at 16:06

I teach the students to use parentheses only. They are widely acceptable and it is what I prefer to use. One argument against brackets is that some students find it hard to remember how to draw different kinds of brackets and spend inordinate amounts of time shaping their brackets.

However I find that it is important to show students that different brackets may be used as parentheses and are not only used for matrices or the floor function. When students find an expression with different brackets they need to be able to understand it and know how to evaluate correctly.

• I'm used to [ ] being used in place of ( ), but I would never expect to see { } used that way. Oct 4, 2015 at 15:16
• @JessicaB As an upper elementary school teacher I have seen { } used that way, but not often. Students who haven't seen them before are often thrown by them. Oct 7, 2015 at 0:08
• As research mathematician who hasn't seen them before, I would be thrown by them. Oct 7, 2015 at 6:56

I'm in favour of using the different brackets : '(' and ')', '[' and ']' and finally, '{' and '}'.

Why?
Well: $$[2 \cdot (3 + 5)]^6$$ is much clearer than $$(2 \cdot (3 + 5))^6$$, especially for weaker students.

Your reaction: "But computers don't understand this notation!"
My reaction back: "People come before machines!"

When you introduce brackets, you can easily say:

1. First the calculations inside '(' and ')'.
2. Afterwards the calculations inside '[' and ']'.
3. Finally the calculations inside '{' and '}'.

Like this, the students learn about calculating from inner to outer.
Once they have grasped that idea, they might be confronted with machines, who are not even capable of working with different brackets.
"Indeed: a computer is a machine, who is not capable of things which are obvious for you, so you are smarter than a computer!" (never miss an opportunity to praise your students!).

• I have received a downvote on this answer. Why? Jan 13 at 9:31
• @Nij: you think having a point, but you are so much missing the point: this discussion is about teaching students to embed brackets. This is done, using a depth of 2 or maximum 3, and this is generally teached at children of the age of seven tot ten years old, not more. The situation you're describing, with the five, seven of ten brackets is far too high for this age and is only applicable for students who are at least 12 years old. In other words, your comments are not relevant. Jan 14 at 10:16
• @Nij: what age are you talking about? Jan 14 at 15:31
• @Nij If it actually made things more difficult for students you'd have a point, but that seems to be something you and others have pulled out of thin air. I have plenty of books that use ( and [ and I find that it does make it clearer (almost none use { though, which is fine with me.). Where are these supposed weak students who can parse ten layers of parentheses but would fall apart if some of them are brackets? They don't exist. Jan 14 at 15:54
• This is why people don't bother with a comment to explain downvotes.
– Nij
Jan 14 at 19:27

I would recommend normal parenthesis most of the time, but if it gets confusing, consider:

1. Use smaller parenthesis inside and larger ones outside, matching the sizes of matching left and right parenthesis. Handwriting and LaTeX can both do this.
2. Maybe have the outermost as square brackets, if the above is not sufficient.
3. Also consider splitting the expression and using letters; in mathematical analysis papers, a long integral or other expression might be split up as I + II + III, and then each of those parts analyzed separately. This adds to the readability of a given expression.

There are many ways of making readable nested expression, and I fear that if several different kinds of parenthesis are really useful, the thing is quite pathological (only research folk are likely to meet it), an example to prove a point or poorly organized and could be split or otherwise handled better.

• Using letters in the middle of an arithmetic expression typically comes many years later than using parentheses in the middle of an arithmetic expression, so hopefully by the time the students come across letters, they already know how to deal with parentheses.
– Stef
Jan 17 at 14:18

Your experience at school flys in the face of convention. Teaching your students that way will not prepare them well for meeting the rest of the world

This is incorrect. Brackets and braces as second and third layers of parentheses are still widely used in "the rest of the world".

For example, the APA Style Guide prescribes

When a mathematical equation contains one level of enclosure, use parentheses, ( ); for two levels, add brackets outside, [( )]; for three levels, add curly brackets outside, {[( )]}.

Likewise, the Chicago Manual of Style (2017, 7e, 12.26) prescribes

One can argue that this is illogical or not conforming with current practice of most mathematicians. But this is clearly still widely used.

So my recommendation would be to teach both.

• This is interesting! I rarely see curly brackets so I wouldn't expect to see them recommend as some sort of standard. Just more proof that students need to be exposed to all sorts of notation. Jan 16 at 16:13

To answer your question, no it is not wrong to suggest that they use all parentheses if that's what you prefer, but I don't think it's a good idea to be too strict about it if they find it easier to parse with some combination of parentheses and brackets. If they prefer something like $$[\sin(x+y)+\sin(x-y)]$$ to $$\left(\sin(x+y)+\sin(x-y)\right)$$ then they're in good company. I certainly prefer the former.

I can't emphasize enough how much it is not a convention to only use parentheses. I have no idea where people are getting that! I flipped through a variety of books in my house and here are some that use $$($$ and $$[$$, and in fact even some $$\{$$.

1-5) The calculus books of Stewart, Thomas/Finney, Kline, Spivak, and Openstax, 6) Physics 6e, Serway/Jewett, 7) Electricity and Magnetism 3e, Purcell/Morin, 8) Classical Electrodynamics 3e, Jackson, 9) Modern Classical Physics, Thorne/Blandford, 10) Visual Differential Geometry and Forms, Needham, 11) Applied Longitudinal Analysis 2e, Fitzmaurice/Laird/Ware, 12) Advanced Engineering Mathematics 10e, Kreyszig, 13) Mathematics and Its History 3e, Stillwell 14) Strang's Linear Algebra and Its Applications 4e

Only about 1 in 3 or 1 in 4 books I checked used only parentheses, so obviously it's to their benefit to at least be exposed to square brackets at some point in their education.

The use of the full range of options for enclosing sections of equations is still alive and well in the scientific world. As one recent example, the first article I pulled from the December 22 issue of Physical Review contains the following equations:

The hierarchy is clearly shown. There is a reason they were put into TeX by Knuth eons ago.