How to teach brackets?

Question

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 $(((3+4)\cdot 4)^4)^{1/2}$

Answer 1

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.

Answer 2

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.

Answer 3

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!).

Answer 4

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.

Answer 5

The accepted answer claims that

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.

Answer 6

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.

Answer 7

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.