THE TRUTH
PEDM
ED
A hierarchy-of-operators
triangle shapes students’
conceptual understanding
of the order of operations.
414
MatheMatics teaching in the Middle school
●
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Vol. 16, No. 7, March 2011
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“The other convention involves order
of operations. Multiplication and division are always done before addition
and subtraction.” (van de Walle and
Folk 2005, p. 425)
When learning the order of operations, students are instructed to
adhere to the directive above when
determining the numerical value of an
arithmetic expression. A more typical
approach is the use of a popular mnemonic called PEDMAS (parentheses,
exponents, division, multiplication,
addition, subtraction) or BEDMAS
(brackets and so on).
ABOUT
MAS
BeTSy DUPUIS/ISTOCKPHOTO.COM
Jerry A. Ameis
The literature is scant on conceptual approaches to teaching order of
operations. The approaches do not
involve students developing a significant understanding of why the
indicated order should be followed or
under what conditions it should be
followed. The emphasis is on students
following PEDMAS (or variants) in a
prescribed manner.
Rambhia’s (2002) article contains
an example. A word problem is briefly
discussed that can help students appreciate why multiplication seems to
have priority over addition, but the article does not provide depth or answer
this question: Does one always have
to do all the multiplication before
any addition can be done? The article
mentions these four levels of order:
“multiplicative” clumps of an expression, such as
Level 1: { [ ( ) ] }
Level 2: Exponents
Level 3: Multiplication and division
Level 4: Addition and subtraction
to simplify and break down the
expression into more manageable
pieces. However, in the final analysis,
students are instructed to follow the
hierarchy of the levels without any
development of why they should do
so or whether alternative processing
is also possible (and valid). Rambhia
expresses the core instructional intent
this way: “Organizing the order of
operations into a table format allows
Although these levels could be
promising in terms of developing a
conceptual understanding of the order
of operations, the article does not
explain the existence of those levels.
It mentions students identifying
Vol. 16, No. 7, March 2011
●
2 × 3(4 + 5),
MatheMatics teaching in the Middle school
415
Fig. 1 Incorrect applications of the order of operations provide evidence of the need for
conceptual understanding of the order of operations.
students to remember and apply the
hierarchical rules of order of operation more proficiently than traditional
methods” (2002, p. 195).
pedMas and
stUdent errors
PEDMAS and its variants do not encourage conceptual understanding of
the order of operations. They may also
partially explain why students make
particular mistakes in arithmetic and
in corresponding algebraic forms. Two
examples follow.
Figure 1a shows how students
might incorrectly obtain a value for
the expression
(a)
25 + 16
.
5+ 4
(b)
Fig. 2 A non-PeDMAS evaluation (a) of a complex expression produces the correct
results (b) in an “aha!” moment.
2 + 3 × 5 + 7 + 4 × (3 + 2 × 5 + 1) + 10 + 2 × 3 − 2
2 + 3 × 5 + 7 + 4 × (3 + 2 × 5 + 1) + 8 + 2 × 3
17 + 3 × 5 + 4 × (3 + 2 × 5 + 1) + 2 × 3
17 + 15 + 4 × (3 + 2 × 5 + 1) + 6
38 + 4 × (3 + 2 × 5 + 1)
38 + 4 × (4 + 2 × 5)
38 + 4 × (4 + 10)
38 + 4 × (14)
38 + 56
94
(a)
2 + 3 × 5 + 7 + 4 × (3 + 2 × 5 + 1) + 10 + 2 × 3 − 2
2 + 3 × 5 + 7 + 4 × (3 + 10 + 1) + 10 + 2 × 3 − 2
2 + 3 × 5 + 7 + 4 × (14) + 10 + 2 × 3 − 2
2 + 15 + 7 + 56 + 10 + 6 − 2
17 + 7 + 56 + 10 + 6 − 2
24 + 56 + 10 + 6 − 2
80 + 10 + 6 − 2
90 + 6 – 2
96 – 2
94
(b)
416
MatheMatics teaching in the Middle school
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Vol. 16, No. 7, March 2011
One explanation for this error is that
PEDMAS states that division must be
done before addition. Figure 1b illustrates how students might incorrectly
solve 59 + 4. One explanation is that
PEDMAS states that exponentiation
must be done before addition. These
two incorrect solutions show that the
division bar and the radical symbol are
confusing to students. It is important
to emphasize to students that these
items are considered grouping symbols
and are to be considered in the same
way as a bracket or a parenthesis, in
addition to the operation they imply.
the MYth oF pedMas
as a MatheMatical law
The K−8 preservice teachers that I
teach typically view PEDMAS as a
mathematical law because of the way
they were taught. This viewpoint
should be reconstructed if they are to
become more effective in developing
student understanding of the order of
operations.
To begin, I ask the preservice
teachers to provide several arithmetic
expressions of varying complexity,
then obtain values by applying
PEDMAS backward (subtraction,
addition, and so on).
After I obtain a value for an expression by applying a non-PEDMAS sequence of calculations (see fig. 2), the
preservice teachers work out a value in
the normal PEDMAS manner. They
are amazed that the two values are the
same. Thus begins the journey into deconstructing the PEDMAS myth and
establishing a conceptual understanding of the order of operations.
The approach used to develop a
conceptual understanding involves
two overlapping aspects:
1. A hierarchy-of-operators triangle,
and
2. Any-which-way processing (leftto-right processing is not a mathematical law either).
The development of these two aspects allows the preservice teachers to
experience learning through problem
solving (unfortunately, an experience
distinctly different from how they
learned mathematics). These aspects
also give examples of ways to develop
a conceptual understanding of the
order of operations.
To help develop the hierarchy-ofoperators triangle and any-which-way
processing, I tell Rocky the Squirrel
stories. Although intended for adults
and somewhat corny, they illustrate
how a story can be used to help teach
mathematics.
Left-to-right
processing is
not a
mathematical
law.
wife. Today was no different. She
said, using a most sarcastic tone,
“So, waiting for the acorns to grow
into trees, are you?” Rocky knew
that tone well. He scrambled to
his feet and began gathering and
storing acorns. He put 7 acorns in
a hole under the beech tree, followed by 2 piles of 6 acorns each in
a split in the old cottonwood tree.
He proudly told his wife that he
was hard at work. She asked, “How
many acorns have you stored?”
To answer the question in the story,
I first ask my students to represent
Rocky’s efforts with an arithmetic
expression. Responses usually include
7 + 2 × 6, 2 × 6 + 7, and 6 + 6 + 7. I
then ask which expression most accurately represents the time sequence of
events and the exact way that Rocky
stored the acorns. We eventually agree
that the expression 7 + 2 × 6 best represents the activity in the story.
I ask students to work out an
answer to the expression. About half
of them add 7 and 2 and multiply the
sum by 6, obtaining a result of 54.
Others multiply 2 by 6 and then add
7 to the product, obtaining 19 as the
number of acorns. I do not indicate
right or wrong at this point.
I then ask a student to draw a
picture of the situation on the white
board (see fig. 3). After the students
have agreed that the drawing is appropriate, I ask them to count the
number of stored acorns indicated
by the picture. They count 19. This
result is compared with the two results
obtained previously. The students
conclude that 7 + 2 × 6 must equal
19, because that result agrees with the
total in the drawing.
We discuss which operation has
priority and when. The crucial matter
developed through our discussion is
Fig. 3 A pictorial representation of rocky’s acorn gathering provides insight into applying
an order of operations to 7 + 2 × 6.
the addition and
MUltiplication sitUation
I begin with a story that involves an
addition and multiplication situation.
The story is also intended to provide
an example of where left-to-right
processing is incorrect.
Fall was coming. Rocky was a
“leave it for another day” kind of
squirrel. He did things only when
pushed into action, usually by his
Vol. 16, No. 7, March 2011
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MatheMatics teaching in the Middle school
417
erwise, addition can be done before
division. The deeper understanding
being developed (and in the addition
and multiplication situation beforehand) is that priority of operation
depends on the particulars of a situation rather than on the order dictated
by PEDMAS or the hierarchy-ofoperators triangle. For example, even
in a simple expression, such as
Jerry A. Ameis
Producing a visual of
the order of operations
illustrates which
operations have priority.
that the number 2 in the 7 + 2 × 6
expression is faced with a choice. It
can belong to the “7 +” or the “× 6.”
When that choice occurs in an expression, multiplication has priority over
addition. We obtain values for more
complex expressions to strengthen
the understanding that multiplication
has priority over addition only when a
choice between the two operations is
involved.
I conclude by drawing a triangle
and calling it the hierarchy-of-operators
triangle (an appropriate label for
middle-grades students could be the
boss triangle, because one operation is
a boss over the other), and then record
the first conclusion (see fig. 4a).
To deconstruct further the simplistic PEDMAS thinking that
multiplication is done first, no matter
what, I tell another Rocky story. The
resulting expression involves more
than one addition (for example,
2 + 5 × 4 + 3). Once again, I ask
students to draw a picture of the
situation; they obtain an answer of 25
acorns. We discuss various ways of
processing 2 + 5 × 4 + 3 to help them
realize that the numbers 2 and 3 can
be added before, but separately from,
the multiplication of 5 × 4. This
reinforces the important point that
multiplication is “boss” over addition,
only when a number has the choice of
belonging to “add” or belonging to
418
“multiply.” The various ways of
obtaining an answer for the expression
(e.g., 4 × 5, then 2 + 3; or 2 + 3, then
4 × 5) help promote an understanding
of any-which-way processing.
2 + 12 ÷ 4 + 5 + 7,
one does not have to do the division
before the addition. One can calculate
2 + 5 + 7, then add that result to
12 ÷ 4.
The Subtraction with
Multiplication or Division
SITUATION
We next discuss the situations of
multiplication and subtraction and
of division and subtraction. The
preservice teachers see addition and
subtraction as similar operations. For
this reason, they have little difficulty
concluding that multiplication has
priority over subtraction and division
has priority over subtraction when the
choice of “belonging to” occurs. All
these conclusions are recorded on the
hierarchy-of-operators triangle (see
fig. 4b).
THE Addition AND Division
SITUATION
Rocky stories are also employed when
addition and division are considered.
Once again, students draw pictures,
count the acorns, and compare the
picture count with the results obtained
by evaluating the expressions that
represent the stories. The conclusion
we reach is that division takes priority
over addition only when a number appears to have the choice of belonging
to “divide” or belonging to “add.” Oth-
Fig. 4 In (a), a multiplication and addition hierarchy shows who is boss. Shared boss
duties are highlighted in the addition, multiplication, division, and subtraction hierarchy
shown in (b).
Mathematics Teaching in the Middle School
(a)
●
Vol. 16, No. 7, March 2011
(b)
eQUal prioritY For
MUltiplication and division
The matter of multiplication and
division having equal priority is problematic for two reasons:
1. Rocky the Squirrel stories do not
naturally lend themselves to arithmetic expressions like 12 ÷ 2 × 3.
(For example, “Rocky placed 12
acorns into 2 equal piles and then
tripled each pile. How many acorns
are in each pile?” is awkward.)
2. The proper thing to do appears to
be left-to-right processing.
Rather than employing lessthan-desirable Rocky stories, I
change division notation to fraction notation in a multiplication or
division expression and discuss various ways of processing the transformed expression. For example, the
expression
30 ÷ 2 × 15 ÷ 3 × 4
becomes
30 ×
1
1
× 15 ×
× 4.
2
3
When the expression is written using fraction notation, the preservice
teachers more readily conclude that
multiplication and division have equal
priority. However, for this approach
to make sense, they must understand
fraction multiplication.
topping the hierarchY-oFoperators triangle
At this point, many preservice teachers wonder why the top of the hierarchy-of-operators triangle is vacant.
Their PEDMAS background propels
them to think that brackets belong
there, but I discourage this thinking
by asking two questions:
1. What is in the triangle so far?
2. Are brackets arithmetic operations?
We then discuss brackets as being
containers rather than as indicators of
processing order. We obtain values for
expressions such as
BeTSy DUPUIS/ISTOCKPHOTO.COM
eQUal prioritY For
addition and sUbtraction
Not surprisingly, most preservice
teachers challenge the pairings of
+ and – and of × and ÷ at the same
level. PEDMAS and their past mathematics learning experiences have led
them to believe that addition and subtraction or multiplication and division
do not have equal priority.
Deconstructing the myth that
addition has higher priority than
subtraction is fairly simple. Telling a
story, such as that “Rocky buried
3 acorns, his son ate 2, Rocky hid
7 more acorns, the neighbor stole 1,
and so on,” provides a context for
arithmetic expressions and drawings.
The teachers then realize that addition
and subtraction have equal priority.
In relation to this, one small misconception sometimes emerges during
any-which-way processing of arithmetic
expressions. Some preservice teachers view an addition and subtraction
symbol as rooted in place, rather than
considering that it is attached to the
number it precedes. In 5 − 3 + 4, some
switch the 3 with the 4 (obtaining
5 − 4 + 3), rather than switching the
+ 4 with the – 3 (obtaining 5 + 4 – 3).
This misconception is addressed by
reconnecting the story to the arithmetic
expression that represents it (e.g., Did
Rocky eat 4 acorns or 3 acorns?).
2 × (3 + 4 × 5) + 1 + 8 + 6 × 2
in a variety of ways (sometimes leaving the bracket processing for last) to
help establish that understanding.
I use this story to complete the
hierarchy-of-operators triangle:
Rocky decided that it was time
to move to new quarters, but the
current state of his home would not
fetch many acorns on the real estate
market. Some home improvements
were needed. The front walk was a
good place to begin. Rocky wanted
to make a new walkway consisting
of 3 large squares, with each square
being 4 paving stones across. How
many paving stones will Rocky
need to buy for his new walk?
I ask the preservice teachers to represent the story with an arithmetic
expression. Various ideas involving addition and multiplication are provided
(e.g., 3 × 4 × 4; 16 + 16 + 16). I place
a restriction on the expression; it must
involve a power. After discussion, we
agree on the expression 3 × 42.
As before, I ask someone to draw
a picture (see fig. 5) and count the
number of paving stones indicated,
which is 48. This result is compared
Fig. 5 This illustration representing rocky’s paving stones helps establish the order of
operations for 3 × 42.
Vol. 16, No. 7, March 2011
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MatheMatics teaching in the Middle school
419
The top of the triangle gets
filled in with “powers.”
Jerry A. Ameis
order?—are not addressed by instructing students to apply PEDMAS
dogmatically.
An advantage of developing the
hierarchy-of-operators triangle is that
students can understand—
with different ways of evaluating
3 × 42 (e.g., 122 = 144, and 3 × 16
= 48). We discuss the matter of the
number 4 (it can choose to belong
with the 3 or the exponent 2). The
preservice teachers begin to understand that exponents have priority
when there is a choice between exponentiation and another arithmetic operation. Further examples strengthen
this understanding.
I then write “powers” at the top of
the triangle (see fig. 6). This ends our
discussion with respect to the middlegrades curriculum expectations. However, I mention that mathematical notions
such as sine and square root also belong
in the top part and that, in general,
unary operators (that operate on one
number only) belong in the top part.
To conclude the development of
the order of operations, I ask my students to provide a variety of complex
Fig. 6 A completed hierarchy-of-operators
triangle shows the order of priority,
including the role of powers or exponents.
420
arithmetic expressions and obtain
values for them in at least two ways,
paying attention to the hierarchyof-operators triangle. We compare
results. Unless errors have been made
(e.g., arithmetic errors), the results of
obtaining values in different ways are
the same. This strengthens their functional understanding of the order of
operations in the sense that applying
the hierarchy-of-operators triangle is
only required when confronted with
a conflicting belongs-to choice (e.g.,
+ or ×) in an expression.
Concluding Discussion
The order of operations is an important prealgebra concept that requires
understanding rather than mindless
mimicry (Ameis 1992). The algebra
of real numbers is a generalization
of arithmetic. Without a conceptual
grounding in arithmetic (in this case,
the order of operations), learning and
understanding algebra can become
a difficult road for too many middle
school students. The road can easily
turn into an unsatisfying journey that
consists of memorizing what to do
when one is faced with a variety of
circumstances.
Unfortunately, having students
follow PEDMAS or its variants in a
prescriptive manner does not support a conceptual understanding of
the order of operations. Important
questions —Why is that order valid?
and Do we always have to follow that
Mathematics Teaching in the Middle School
●
Vol. 16, No. 7, March 2011
1. why the order of operations is
valid,
2. under what conditions one needs
to apply the hierarchy,
3. that left-to-right processing is not
a mathematical truth,
4. the various ways to process that are
possible, and
5. that decision making is required
when determining processing
order.
In short, developing the hierarchyof-operators triangle and its application encourages conceptual thinking and understanding. It also has a
potentially significant payoff when
learning algebra. This is not the case
when PEDMAS (or its variants) is
taught to students simply as a rule
to follow.
REFERENCES
Ameis, Jerry. The Validation of a Model
of Pre-cursors of Algebra. PhD diss.,
University of Manitoba, 1992.
Rambhia, Sanjay. “A New Approach to an
Old Order.” Mathematics Teaching in
the Middle School 8 (2002): 193−95.
Van de Walle, John A., and Sandra Folk.
Elementary and Middle School
Mathematics. Canadian ed.
Toronto: Pearson Education, 2005.
Jerry A. Ameis, j.ameis@
uwinnipeg.ca, teaches
mathematics and mathematics methods courses
for K−8 preservice teachers in an access program at the University
of Winnipeg in Manitoba. His research
interests concern how students learn
mathematics and learning issues related
to teaching mathematics.
the
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