How to explain brackets to young students

Question

It's amazing how simple questions from young students can often uncover unexpected gaps in (at least my) knowledge, or at least the ability to explain.

Today, a student asked me why she can "forget the bracket": $$ x+(x+5)=x+x+5 $$ Elementary school student's idea of brackets is I have to calculate this before anything else and thus the student thinks that perhaps $(x+5)$ is a entity of its own, not to be touched (since you can't really add $x$ and $5$).

My approach was to

1. Demonstrate on natural numbers (i.e. proof by example) that no amount of bracketing will change the result with addition to deal with this specific example.

2. Explain that $(x+5)$ and $-(x+5)$ can be thought of as a special case/shorthand of $c(x+5)$ (because multiplying a bracket by a number is something the student's mind automatically recognizes and knows how to do) and thus $(x+5)$ "really" equals $1(x+5)$ and $-(x+5)$ "really" equals $-1(x+5)$, hopefully ensuring the student wont make a mistake in the future.

However, I am not convinced that I succeeded fully at providing a good mental process for dealing with brackets in her mind. Thus I am asking:

How do/would you explain brackets? What is the best/generally accepted way (if there is one)?

Answer 1

I agree with Surb's comment that $(x+5)=1(x+5)$ and $-(x+5)=-1(x+5)$ gives the student something to cling on to. Also, this principle appears time and again when finding the slope of lines $y=\pm x+b=\pm 1x+b$ or in quadratic expressions $f(x)=\pm x^2+bx+c=\pm 1x^2+bx+c$ etc.

So $1,-1$ and $0$ are hidden numbers. Regarding $0$, we have for instance $y=3$ where the student searches for the slope in vain. Eventually the student guesses $a=b=3$, which is wrong, and then the storytelling about Surb's magic numbers begins, in order to tell why $0x=0=\text{nothing}$ is hidden in that expression ;)

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Adressing the comment by cobaltduck to Steve Jessop's answer, suggesting (by quoting Dumbledore +1 for that) that by not calling this rule by the name of associativity just produces fear of what that name represents:

I believe very much in the perspective brought forth in the article "On The Dual Nature of Mathematical Conceptions" by Anna Sfard (published in Educational Studies in Mathematics 22, 1-36, 1991), that the learning process of a student learning mathematics mimics the principles of the learning process of the human race as such in the historical development of the subject.

If a student is at a stage still uncertain whether $x+(x+5)=x+x+5$ the foundations may not yet be laid to put forward the more abstract idea of associativity. I certainly knew $x+(x+5)=x+x+5$ well before I ever heard that terminology. When I first encountered the concept associativity, it very much generalized an intuition/knowledge I already had about manipulating algebraic and numerical expressions.

Whereas a computer can be programmed to follow a set of rules, thus handling those to produce results deductively, I believe that humans learn rules inductively. Having learnt a rule, a human may then apply that rule deductively to new problems within the scope of that rule.

At least that is what I believe.

Answer 2

In my opinion this more a pedagogical that a mathematical problem.

In my old experience of teacher I faced this problem in this way.

1) The parentheses are used to determine the order in which to run the operations. So not always the parentheses are necessary, i.e. if changing the order of operations the result is the same parentheses are unnecessary.

2) Presentation of the properties of operations that requires the use of parentheses or not. In particular associativity (the parentheses can be eliminated) and distributivity (here the parentheses are important).

3) Study of particular cases in which the student has to decide if parentheses are necessary or not, but, in that first time, without subtle use of the neutral elements $1,0$ and of inverse or opposite elements. E.g.: $$ 3+(2+5);\quad (3\cdot2)+5 ;\quad 3\cdot(2+5) ;\quad (3\cdot2)+(5\cdot4);\quad 3\cdot(2+5)\cdot4 $$ and use these to establish the precedence rule of multiplication with respect to addition.

4) verify what of the properties in 2) has a rule in the exercices 3).

5) Introduce, with care, inverse and opposite elements. In my experience this is the most difficult step and it is important that this is done by a series of exercices in which students have to explicit this neutral elements in many different expressions as: $$ \quad 3\cdot[-(1-3)-2)] ;\quad 1-2\cdot[-(\dfrac{1}{2} -1)];\quad-(1-2)\cdot[-(\dfrac{1}{2} -1)] $$ I am convinced that this is not a waste of time.

6) Now we can go to exercices as 3) but with ''hidden'' neutral elements.

Answer 3

Parentheses dictate the order of operations when evaluating an expression. When we say:

2 * 3 + 4

we have to decide in what order those are evaluated, and that choice is pretty much arbitrary. So, we could all agree that

2 * 3 + 4 = 6 + 4 = 10

or that

2 * 3 + 4 = 2 * 7 = 14

It just so happens that everybody has agreed to evaluate multiplication and division before addition and subtraction. So, if we want to have

2 * 3 + 4 = 14

then we need to add parentheses so that it evaluates in the way we want. Specifically, we write

2 * (3 + 4) = 14

and everyone agrees to evaluate the parenthetical subexpression first (PEDMAS). That way, everyone gets the same answer when they evaluate the whole expression.

In the case of

1 + 2 + 3 = 6

it just so happens that, no matter where you put the parentheses, you always get 6.

1 + 2 + 3 = 1 + (2 + 3) = (1 + 2) + 3 = (1 + 2 + 3) = ((((1 + ((2) + 3)))))

And if you always get the same number, then the expressions are equal, and you can say so. And so you end up "ignoring the brackets".

In the case of

x + (x + 5) = x + x + 5

is there any value of x where those wouldn't end up being equal?

Here's another example:

x + (((x + 5))) = x + (x + 5)

(Anyway, that's my shot at explaining $f = g \iff \forall x \in \mathbb{R}, f(x) = g(x)$)

Answer 4

Perhaps it's worth taking a step back and reminding your students how they learned to do addition.

If I have a pile of X jellybeans, and another pile of X jellybeans, and another pile of 5 jellybeans, does it matter which order I put them together?

hmmm... jellybeans :)

Edit: in response to comment, please feel free to replace jellybeans with an alternative confection of your choice.

Answer 5

I assume if you're using $x$, then you're at the point of teaching that we manipulate expressions according to rules. We have a rule here, called "forgetting the brackets", and the task is to justify it both in terms of why it's good mathematics, and also because the student can remember and use things that make sense much more easily than a long list of apparently-arbitrary information.

I certainly agree that using some examples to demonstrate that it works is a good first step. If this satisfies the student, OK, but we want them to understand the rule and not merely accept it.

So, similar to Zaaier, I would argue that the reason we're allowed to write:

$$x + x + 5$$

is that:

$$(x + (x + 5)) = ((x + x) + 5)$$

This property of addition is called "associativity", although considering properties of general operators might be too much abstraction. Judge this for the individual student.

Anyway, this fact about addition is what permits us to "forget the brackets" anywhere in a sequence of additions. Bring commutativity into it too, and you can also reorder the summands:

$$x + 5 + x = (x + 5) + x$$

(the implied meaning)

$$= x + (x + 5)$$

(by commutativity)

$$= (x + x) + 5$$

(by associativity)

Of course, I've intentionally ignored something. The next, perfectly reasonable, question you'll face is why we are allowed to write:

$$x - x - 5$$

despite that:

$$((x - x) - 5) \neq (x - (x - 5))$$

The reason for this is that we're applying a "left to right" rule, so that in the absence of brackets $x - x - 5$ always means $((x - x) - 5)$, not $(x - (x - 5))$. This is the reason why we cannot "forget the brackets" in $x - (x - 5)$, even though we can forget them in $x + (x + 5)$.

Similarly we have precedence rules that allow us to write $2x + 5$ to mean $(2x) + 5$ and so on. Also be careful if you do sling around big words like "associativity". Addition is associative and multiplication is associative, but you need to teach that this doesn't imply that:

$$(x + y) \times z = x + (y \times z)$$

The crucial point is that we could always put brackets everywhere, but we don't want to have to write them, so we've invented rules to keep the number of brackets down while ensuring that what we write has a single possible meaning. "Forgetting the brackets" is what we do when (and only when) the shorter expression without brackets either means exactly the same thing as the longer one with brackets, or else is equal because the order of calculation doesn't matter.

Answer 6

I think compare and contrast is a good approach here. As you have demonstrated that no amount of re-arranging parentheses changes the outcome for addition, I would then contrast this with something that is not associative, like division. My favorite example is

$$8\div4\div2 = ?$$

which is either

$$(8\div4)\div2 = 2\div2 = 1$$

or

$$8\div(4\div2) = 8\div2 = 4.$$

I think this shows the real power of associativity and why fractions are so difficult to learn.

As an aside, while in grad school I had a gentleman "prove" to me that the units of force ($kg\ m/s^2$) are identical to energy ($kg\ m^2/s^2$) because he treated division as if it were associative.

Answer 7

There are parenthesis, (), and there are brackets, []. These are used the same way in mathematics*, but usually parenthesis are used first, and then when there are too many parenthesis groups, brackets are used to group those together - but for the most part, parenthesis and brackets are the same tool.

To answer your student's question, why can we "forget" the bracket? That is a wonderful way to put it, we can forget the parenthesis because [in the example you gave] they are meaningless mathematically. Something to bring up with your students to start things off -

Parenthesis (or brackets) are used two different ways in mathematics:

1) to visually (but arbitrarily) separate a group of symbols, so as to make them easier to see, read, and think about... kind of like saying, "hey, you really should group these together in your thinking, but you don't have to."

2) to dictate order of operation, so that there is no ambiguity as to what to do first, second, ..., last.

Your example is the number one category, so the parenthesis can be "forgotten", ignored, not used at all, it is all the same, but perhaps the mathematician wishes to emphasize the (x + 5), and so they place it in parenthesis to express that group. But without the parenthesis, the expression is the same.

Regarding example two - consider the following expressions:

a) 2 + 4 * 6 = 48

b) 2 + (4 * 6) = 48

c) (2 + 4) * 6 = 36

For example a, without parenthesis it isn't so clear what to do first, but the order of operations dictates that multiplication occurs first, and then addition, and so the expression is equal to 48. But this is easier to see with example b, and so example b is the typical way to write the expression, even though we can "forget" the parenthesis if we wanted to. However, make sure your student understands that forgetting the parenthesis in example c isn't okay, because with the parenthesis the expression is equal to 36, but without the parenthesis, the expression is equal to 48. This is because the order of operations requires that parenthesis occur before anything else. And so in this way, we can forget the parenthesis for expressions like a and b, where the parenthesis are meaningless in that the expression is computed in the same order of operation as it would have been without parenthesis, but we can not forget the parenthesis when they are dictating the order of operation to use, so that otherwise the parenthesis-less expression would equal something else.

And yes - it is amazing what those kids can get you to think about.

• This isn't true for advanced mathematics.

Answer 8

I believe that even very young students can comprehend the necissity of the associative law in relation to binary operations( +, -, *, /). Which is where to start, no matter what.