Maria Hernandez, NCSSM/Amplify learning.
Angela Teachey, NCSSM
hernandez@ncssm.edu and teachey@ncssm.edu
Instructions: Let 𝑓 𝑥 = 𝑥 +
1
.
2𝑥
Calculate 𝑓 4 .
Student’s Question: For the fraction part, do I
plug in 4 and leave it in the denominator, or do
1
I multiply 4 by ?
2
My response: That should be apparent from
the notation. I can’t answer your question.
Student: It depends on who you ask.



CCSS.Math.Content.5.OA.A.1 Use parentheses,
brackets, or braces in numerical expressions, and
evaluate expressions with these symbols.
CCSS.Math.Content.5.OA.A.2 Write simple
expressions that record calculations with
numbers, and interpret numerical expressions
without evaluating them. For example, express
the calculation “add 8 and 7, then multiply by 2”
as 2 × (8 + 7). Recognize that 3 × (18932 + 921)
is three times as large as 18932 + 921, without
having to calculate the indicated sum or product.
CCSS.Math.Content.6.EE.A.2c Evaluate expressions at
specific values of their variables. Include expressions
that arise from formulas used in real-world
problems. Perform arithmetic operations, including
those involving whole-number exponents, in the
conventional order when there are no parentheses to
specify a particular order (Order of Operations). For
example, use the formulas V = s3 and A = 6 s2 to
find the volume and surface area of a cube with sides
of length s = 1/2.
1.
More complex grouping symbols?
2.
Students struggle to recognize and understand
structure in algebraic expressions?
3.
Students must construct expressions with correct
order of operations rather than just interpreting
given expressions?
4.
Distributive property a confounding factor in the
transition to algebraic structures and functions?
Algebra: Seeing Structure in Expressions
 CCSS.Math.Content.HSA-SSE.A.1 Interpret
expressions that represent a quantity in terms of
its context.★
◦ CCSS.Math.Content.HSA-SSE.A.1a Interpret parts of an
expression, such as terms, factors, and coefficients.
◦ CCSS.Math.Content.HSA-SSE.A.1b Interpret complicated
expressions by viewing one or more of their parts as a
single entity.

CCSS.Math.Content.HSA-SSE.B.3 Choose and
produce an equivalent form of an expression to
reveal and explain properties of the quantity
represented by the expression.
Look for and make use of structure
Mathematically proficient students look closely to
discern a pattern or structure. …Later, students will see
7 × 8 equals the well remembered 7 × 5 + 7 × 3, in
preparation for learning about the distributive
property.
They can see complicated things, such as some
algebraic expressions, as single objects or as being
composed of several objects. For example, they can see
5 – 3(x – y)2 as 5 minus a positive number times a
square and use that to realize that its value cannot be
more than 5 for any real numbers x and y.
1. More complex grouping symbols?
◦ New types of notation:





radical signs
exponentiation notation
fraction bars
absolute value symbols
arguments of trigonometric and logarithmic expressions
◦ Memory aids like PEMDAS and “Please Excuse My
Dear Aunt Sally” not easily transferable to all
algebraic grouping symbols



“… unfortunately, the rules have serious
drawbacks”
“… the criterion of addition before
subtraction still does not allow us to solve 5 8+4 correctly.
“Finally, use of a rule as complicated and
arbitrary as these rules tend to become,
involves far too much of the flavor of rote
learning which has made arithmetic so
unsavory in the past”
Source: Bender, M.. (1962). Order of Operations in Elementary
Arithmetic.The Arithmetic Teacher, 9(5), pp. 263-267.
2. Students struggle to recognize and
understand structure?
◦ Is there a cognitive gap when students transition
from operating on numbers to operating on
unknowns (variables) ?

“… it seems that many teachers and textbook
authors are unaware of the serious cognitive
difficulties involved in the learning of algebra. As
a result, many students do not have the time to
construct a good intuitive basis for the ideas of
algebra or to connect these with the prealgebraic ideas they have developed in [earlier
coursework]; they fail to construct meaning for
the new symbolism and are reduced to
performing meaningless operations on symbols
they do not understand.”
Herscovics, N., & Linchevski, L. (1994). A Cognitive Gap between
Arithmetic and Algebra. Educational Studies in Mathematics, 27(1),
pp. 59-78.
3. Students must construct expressions with correct
order of operations rather than just interpreting
given expressions?



At the algebra, precalculus, and calculus levels,
students must often construct expressions that
convey the correct order of operations, requiring
them to reverse their thinking.
Particularly relevant when entering functions properly
into graphing utilities and other computing software.
Perhaps it is not wise to assume that this is always a
straightforward transition.

An algebra student is using a calculator to
investigate a horizontal compression, 𝑦 =
1
1
2𝑥
, on
the toolkit function 𝑦 = and is confused to see
𝑥
that the transformed reciprocal function is looks
linear.
4. Distributive property a confounding factor in the
transition to algebraic structures and functions?

Distributive property receives heavy emphasis in middle
grades.
◦ CCSS.Math.Content.6.EE.A.3 Apply the properties of operations to
generate equivalent expressions. For example, apply the distributive
property to the expression 3 (2 + x) to produce the equivalent expression
6 + 3x; apply the distributive property to the expression 24x + 18y to
produce the equivalent expression 6 (4x + 3y)

Students may transfer this knowledge inappropriately leading
them to believe that other operations distribute over
addition?
◦ 𝑥+𝑦 =
◦ 𝑥+𝑦 =
◦ log 𝑥 + 𝑦
◦ sin 𝑥 + 𝑦
𝑥+ 𝑦
𝑥 + 𝑦
= log 𝑥 + log 𝑦
= sin 𝑥 + sin(𝑦)
“… many other teachers and researchers identify a … culprit
as responsible for student difficulties with expression
transformation, namely a strong tendency for students to
overgeneralize. Schwartzman (1986) provides examples of
overgeneralization. In an article entitled “The A of a B is the
B of an A,” he writes about how students have difficulty
restraining themselves from overgeneralizing the notion of
the distributivity of one operation over another. He provides
a list of twenty non-equivalencies that students are prone to
1
1
1
assume. These include 𝑎 + 𝑏 𝑛 ≠ 𝑎𝑛 + 𝑏 𝑛 ,
≠ + , and
𝑎+𝑏
𝑎
𝑏
𝑛
𝑛
𝑛
𝑎 + 𝑏 ≠ 𝑎 + 𝑏.
All of these examples are variations on true distributive
statements, such as ‘the power of a product is the product
of the powers’ ... However, they are overgeneralizations in
that the student who accepts them is assuming fewer
constraints on which operations distribute over which others
than actually exist.”
Merlin, E. M. (2008). Beyond PEMDAS: Teaching Students
to Perceive Algebraic Structure (Masters thesis). Retrieved
from Dissertations and Theses database. (UMI No. 5636)

Structure and Grouping Symbols: How do we
prioritize our work?
◦ Work from the “inside out.”

Examples
2∙5+8
◦ When computing an expression such as
, the
6−3
fraction bar has the same function as parentheses.
Any operations that appear above or below a
fraction bar should be completed first.
◦ Compute 0.5 +
9.2−0.2(6)
2
2+ 4 x 5
“Students tend to solve expressions based on
how the items are listed, in a left-to-right
fashion, consistent with their cultural tradition
of reading and writing English. Therefore, the
rules underlying operation order actually
contradict students’ natural way of thinking.”
Welder, R. (2006). Prerequisite Knowledge for
the Learning of Algebra. Hawaii International
Conference on Statistics, Mathematics, and
Related Field.
“However, Kieran suggests that if an
equation such as 3 x 5 =15 were
3 x 3+2 =15, students would realize
that bracket/parentheses usage is
necessary to keep the equation
balanced (Kieran, 1979).”
Welder, R. (2006). Prerequisite Knowledge for
the Learning of Algebra. Hawaii International
Conference on Statistics, Mathematics, and
Related Field.
“They found that students tended to over
generalise the order, usually giving addition
priority over subtraction; or using operations in
left to right order; they can show lack of awareness
of possible internal cancellations; they can see
brackets as merely another way to write
expressions rather than an instruction to act first,
for example:
926 – 167 – 167 and 926 – (167 + 167) yielded
different answers (Nickson, 2000 p. 120); they also
did not understand that signs were somehow
attached to the following number.”
http://www.nuffieldfoundation.org/sites/default/files/P6.pdf
2007 review of research literature on how children learn
mathematics, commissioned by the Nuffield Foundation
From Nephew’s Homework:
 Equality
 Equivalence
 Calculate
 Which
of these do we mean?
“Beginning algebra students tend to see the
equal sign as a procedural marking that tells
them “to do something,” or as a symbol that
separates a problem from its answer, rather
than a symbol of equivalence (Behr, Erlwanger,
& Nichols, 1976, 1980). Even college calculus
students have misconceptions about the true
meaning of the equal sign (Clement, Narode, &
Rosnick, 1981).”
Welder, R. (2006). Prerequisite Knowledge for
the Learning of Algebra. Hawaii International
Conference on Statistics, Mathematics, and
Related Field.




Algebra Touch
Mathination
SymCalc HD (CAS)
Other ideas for how
to use CAS?
From Steve Leinwand’s Accessible Mathematics
“…many students need different modes of
access to mathematical concepts to be
successful.
It should be clear that we will never attain the
important and ambitious goals of algebra for
all specifically, and mathematical power for all
generally, until and unless we change how the
material is taught.”

Frequent imbedding of them mathematics skills in
real-world situations and contexts
Given F = 4(S-65)+10, find F when S = 81.
OR
The speeding fine in Vermont is $4 for every mile per
hour over the 65 mph speed limit, plus a $10 handling
fee.
Then we can build algebraic understanding such as:
 What is the fine if you are caught going 81 mph?
 How fast must have you been going if the fine was
$102?
 Create a graph that shows the relationship between
the speed and the fine.


Learn and reflect – Can help us to understand
each other’s perspective and help us to meet
students where they are.
Share ideas – How do students learn Order of
Operations in middle grades? How do middle
grades teachers use alternatives
representations – geometric models, others?



How can HS teachers “remind” students to
make sense of the algebraic notions based on
their experience with operations of numbers?
How can we help them see when they are
overgeneralizing?
How can we help students see the algebraic
structure in their work?